[ { "id": "authors:wrmhn-d9c80", "collection": "authors", "collection_id": "wrmhn-d9c80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210210-124101903", "type": "article", "title": "Quantum limit cycles and the Rayleigh and van der Pol oscillators", "author": [ { "family_name": "Arosh", "given_name": "Lior Ben", "clpid": "Arosh-Lior-Ben" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-Ron" } ], "abstract": "Self-oscillating systems, described in classical dynamics as limit cycles, are emerging as canonical models for driven dissipative nonequilibrium open quantum systems and as key elements in quantum technology. We consider a family of models that interpolates between the classical textbook examples of the Rayleigh and the van der Pol oscillators and follow their transition from the classical to the quantum domain, while properly formulating their corresponding quantum descriptions. We derive an exact analytical solution for the steady-state quantum dynamics of the simplest of these models, applicable to any bosonic system\u2014whether mechanical, optical, or otherwise\u2014that is coupled to its environment via single-boson and double-boson emission and absorption. Our solution is a generalization to arbitrary temperature of existing solutions for very-low, or zero, temperature, often misattributed to the quantum van der Pol oscillator. We closely explore the classical to quantum transition of the bifurcation to self-oscillations of this oscillator, while noting changes in the dynamics and identifying features that are uniquely quantum.", "doi": "10.1103/PhysRevResearch.3.013130", "issn": "2643-1564", "publisher": "American Physical Society", "publication": "Physical Review Research", "publication_date": "2021-02", "series_number": "1", "volume": "3", "issue": "1", "pages": "Art. No. 013130" }, { "id": "authors:thqk5-es491", "collection": "authors", "collection_id": "thqk5-es491", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190307-131059551", "type": "article", "title": "Exotic states in a simple network of nanoelectromechanical oscillators", "author": [ { "family_name": "Matheny", "given_name": "Matthew H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Emenheiser", "given_name": "Jeffrey", "clpid": "Emenheiser-J" }, { "family_name": "Fon", "given_name": "Warren", "orcid": "0000-0002-5447-2324", "clpid": "Fon-Warren" }, { "family_name": "Chapman", "given_name": "Airlie", "clpid": "Chapman-A" }, { "family_name": "Salova", "given_name": "Anastasya", "clpid": "Salova-A" }, { "family_name": "Rohden", "given_name": "Martin", "clpid": "Rohden-M" }, { "family_name": "Li", "given_name": "Jarvis", "clpid": "Li-Jarvis" }, { "family_name": "Hudoba de Badyn", "given_name": "Matthias", "orcid": "0000-0003-0955-2381", "clpid": "Hudoba-de-Badyn-M" }, { "family_name": "P\u00f3sfai", "given_name": "M\u00e1rton", "clpid": "P\u00f3sfai-M" }, { "family_name": "Duenas-Osorio", "given_name": "Leonardo", "clpid": "Duenas-Osorio-L" }, { "family_name": "Mesbahi", "given_name": "Mehran", "clpid": "Mesbahi-M" }, { "family_name": "Crutchfield", "given_name": "James P.", "orcid": "0000-0003-4466-5410", "clpid": "Crutchfield-J-P" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "D'Souza", "given_name": "Raissa M.", "clpid": "D'Souza-R-M" }, { "family_name": "Roukes", "given_name": "Michael L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "Synchronization of oscillators, a phenomenon found in a wide variety of natural and engineered systems, is typically understood through a reduction to a first-order phase model with simplified dynamics. Here, by exploiting the precision and flexibility of nanoelectromechanical systems, we examined the dynamics of a ring of quasi-sinusoidal oscillators at and beyond first order. Beyond first order, we found exotic states of synchronization with highly complex dynamics, including weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states. Through theory and experiment, we show that these exotic states rely on complex interactions emerging out of networks with simple linear nearest-neighbor coupling. This work provides insight into the dynamical richness of complex systems with weak nonlinearities and local interactions.", "doi": "10.1126/science.aav7932", "issn": "0036-8075", "publisher": "American Association for the Advancement of Science", "publication": "Science", "publication_date": "2019-03-08", "series_number": "6431", "volume": "363", "issue": "6431", "pages": "Art. No. eaav7932" }, { "id": "authors:vzcgf-csb54", "collection": "authors", "collection_id": "vzcgf-csb54", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170911-123703870", "type": "article", "title": "Complex dynamical networks constructed with fully controllable nonlinear nanomechanical oscillators", "author": [ { "family_name": "Fon", "given_name": "Warren", "orcid": "0000-0002-5447-2324", "clpid": "Fon-Warren" }, { "family_name": "Matheny", "given_name": "Matthew H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Li", "given_name": "Jarvis", "clpid": "Li-Jarvis" }, { "family_name": "Krayzman", "given_name": "Lev", "clpid": "Krayzman-L" }, { "family_name": "Cross", "given_name": "Michael C.", "clpid": "Cross-M-C" }, { "family_name": "D'Souza", "given_name": "Raissa M.", "clpid": "D'Souza-R-M" }, { "family_name": "Crutchfield", "given_name": "James P.", "orcid": "0000-0003-4466-5410", "clpid": "Crutchfield-J-P" }, { "family_name": "Roukes", "given_name": "Michael L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "Control of the global parameters of complex networks has been explored experimentally in a variety of contexts. Yet, the more difficult prospect of realizing arbitrary network architectures, especially analog physical networks that provide dynamical control of individual nodes and edges, has remained elusive. Given the vast hierarchy of time scales involved, it also proves challenging to measure a complex network's full internal dynamics. These span from the fastest nodal dynamics to very slow epochs over which emergent global phenomena, including network synchronization and the manifestation of exotic steady states, eventually emerge. Here, we demonstrate an experimental system that satisfies these requirements. It is based upon modular, fully controllable, nonlinear radio frequency nanomechanical oscillators, designed to form the nodes of complex dynamical networks with edges of arbitrary topology. The dynamics of these oscillators and their surrounding network are analog and continuous-valued and can be fully interrogated in real time. They comprise a piezoelectric nanomechanical membrane resonator, which serves as the frequency-determining element within an electrical feedback circuit. This embodiment permits network interconnections entirely within the electrical domain and provides unprecedented node and edge control over a vast region of parameter space. Continuous measurement of the instantaneous amplitudes and phases of every constituent oscillator node are enabled, yielding full and detailed network data without reliance upon statistical quantities. We demonstrate the operation of this platform through the real-time capture of the dynamics of a three-node ring network as it evolves from the uncoupled state to full synchronization.", "doi": "10.1021/acs.nanolett.7b02026", "issn": "1530-6984", "publisher": "American Chemical Society", "publication": "Nano Letters", "publication_date": "2017-10-11", "series_number": "10", "volume": "17", "issue": "10", "pages": "5977-5983" }, { "id": "authors:g1ggs-h1k39", "collection": "authors", "collection_id": "g1ggs-h1k39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180119-133708576", "type": "book_section", "title": "Nonlinear effects in NEMS \u2014 Improving frequency stability", "book_title": "2017 IEEE International Ultrasonics Symposium (IUS)", "author": [ { "family_name": "Villanueva", "given_name": "Luis Guillermo", "clpid": "Villanueva-L-G" }, { "family_name": "Karabalin", "given_name": "Rassul", "clpid": "Karabalin-R-B" }, { "family_name": "Matheny", "given_name": "Mathew H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Cross", "given_name": "Michael", "clpid": "Cross-M-C" }, { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Roukes", "given_name": "Michael Lee", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "Resonant Nanoelectromechanical Systems (NEMS) have generated an enormous interest over the last 15 years, driven by a combination of fundamental questions and practical needs. The combination of relatively high frequencies, high quality factors and small masses make them ideal for a plethora of sensing applications. As an example, it has been possible to detect from a single atom landing on a carbon nanotube till single nuclear spin rotations.", "doi": "10.1109/ULTSYM.2017.8091577", "isbn": "978-1-5386-3384-7", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2017-09" }, { "id": "authors:h3w19-m0r14", "collection": "authors", "collection_id": "h3w19-m0r14", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160429-145516593", "type": "article", "title": "Minimum-action paths for wave-number selection in nonequilibrium systems", "author": [ { "family_name": "Qiao", "given_name": "Liyan", "clpid": "Qiao-Liyan" }, { "family_name": "Zheng", "given_name": "Zhigang", "clpid": "Zheng-Zhigang" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The problem of wave-number selections in nonequilibrium pattern-forming systems in the presence of noise is investigated. The minimum-action method is proposed to study the noise-induced transitions between the different spatiotemporal states by generalizing the traditional theory previously applied in low-dimensional dynamical systems. The scheme is shown as an example in the stabilized Kuramoto-Sivashinsky equation. The present method allows us to conveniently find the unique noise selected state, in contrast to previous work using direct simulations of the stochastic partial differential equation, where the constraints of the simulation only allow a narrow band to be determined.", "doi": "10.1103/PhysRevE.93.042204", "issn": "2470-0045", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2016-04", "series_number": "4", "volume": "93", "issue": "4", "pages": "Art. No. 042204" }, { "id": "authors:yz5mk-5hn90", "collection": "authors", "collection_id": "yz5mk-5hn90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190628-082547756", "type": "monograph", "title": "Frequency Precision of Oscillators Based on High-Q Resonators", "author": [ { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We present a method for analyzing the phase noise of oscillators based on feedback driven high quality factor resonators. Our approach is to derive the phase drift of the oscillator by projecting the stochastic oscillator dynamics onto a slow time scale corresponding physically to the long relaxation time of the resonator. We derive general expressions for the phase drift generated by noise sources in the electronic feedback loop of the oscillator. These are mixed with the signal through the nonlinear amplifier, which makes them {cyclostationary}. We also consider noise sources acting directly on the resonator. The expressions allow us to investigate reducing the oscillator phase noise thereby improving the frequency precision using resonator nonlinearity by tuning to special operating points. We illustrate the approach giving explicit results for a phenomenological amplifier model. We also propose a scheme for measuring the slow feedback noise generated by the feedback components in an open-loop driven configuration in experiment or using circuit simulators, which enables the calculation of the closed-loop oscillator phase noise in practical systems.", "doi": "10.48550/arXiv.1510.07331", "publisher": "arXiv", "publication_date": "2015-10-26" }, { "id": "authors:sqz76-mx572", "collection": "authors", "collection_id": "sqz76-mx572", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150410-095602837", "type": "article", "title": "Building better oscillators using nonlinear dynamics and pattern formation", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Allen", "given_name": "John-Mark A.", "clpid": "Allen-J-M-A" } ], "abstract": "Frequency and time references play an essential role in modern technology and in living systems. The precision of self-sustained oscillations is limited by the effects of noise, which becomes evermore important as the sizes of the devices become smaller. In this paper, we review our recent theoretical results on using nonlinear dynamics and pattern formation to reduce the effects of noise and improve the frequency precision of oscillators, with particular reference to ongoing experiments on oscillators based on nanomechanical resonators. We discuss using resonator nonlinearity, novel oscillator architectures and the synchronization of arrays of oscillators, to improve the frequency precision.", "doi": "10.1007/s12043-014-0924-1", "issn": "0304-4289", "publisher": "Indian Academy of Sciences", "publication": "Pramana", "publication_date": "2015-03", "series_number": "3", "volume": "84", "issue": "3", "pages": "455-471" }, { "id": "authors:h6pk4-31y78", "collection": "authors", "collection_id": "h6pk4-31y78", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140501-094645965", "type": "article", "title": "Eliminating 1/f noise in oscillators", "author": [ { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study 1/f and narrow-bandwidth noise in precision oscillators based on high-quality factor resonators and feedback. The dynamics of such an oscillator are well described by two variables, an amplitude and a phase. In this description we show that low-frequency feedback noise is represented by a single noise vector in phase space. The implication of this is that 1/f and narrow-bandwidth noise can be eliminated by tuning controllable parameters, such as the feedback phase. We present parameter values for which the noise is eliminated and provide specific examples of noise sources for further illustration.", "doi": "10.1103/PhysRevE.89.042901", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2014-04-02", "series_number": "4", "volume": "89", "issue": "4", "pages": "Art. No. 042901" }, { "id": "authors:pajx1-39355", "collection": "authors", "collection_id": "pajx1-39355", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140326-100653458", "type": "article", "title": "Phase Synchronization of Two Anharmonic Nanomechanical Oscillators", "author": [ { "family_name": "Matheny", "given_name": "Matthew H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Grau", "given_name": "Matt", "clpid": "Grau-M" }, { "family_name": "Villanueva", "given_name": "Luis G.", "clpid": "Villanueva-L-G" }, { "family_name": "Karabalin", "given_name": "Rassul B.", "clpid": "Karabalin-R-B" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Roukes", "given_name": "Michael L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "We investigate the synchronization of oscillators based on anharmonic nanoelectromechanical resonators. Our experimental implementation allows unprecedented observation and control of parameters governing the dynamics of synchronization. We find close quantitative agreement between experimental data and theory describing reactively coupled Duffing resonators with fully saturated feedback gain. In the synchronized state we demonstrate a significant reduction in the phase noise of the oscillators, which is key for sensor and clock applications. Our work establishes that oscillator networks constructed from nanomechanical resonators form an ideal laboratory to study synchronization\u2014 given their high-quality factors, small footprint, and ease of cointegration with modern electronic signal processing technologies.", "doi": "10.1103/PhysRevLett.112.014101", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2014-01-10", "series_number": "1", "volume": "112", "issue": "1", "pages": "Art. No. 014101" }, { "id": "authors:ffg80-tnt29", "collection": "authors", "collection_id": "ffg80-tnt29", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140124-082359452", "type": "article", "title": "Phase noise of oscillators with unsaturated amplifiers", "author": [ { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Moehlis", "given_name": "Jeff", "clpid": "Moehlis-J" }, { "family_name": "Wiesenfeld", "given_name": "Kurt", "clpid": "Wiesenfeld-K" } ], "abstract": "We study the role of amplifier saturation in eliminating feedback noise in self-sustained oscillators. We extend previous works that use a saturated amplifier to quench fluctuations in the feedback magnitude, while simultaneously tuning the oscillator to an operating point at which the resonator nonlinearity cancels fluctuations in the feedback phase. We consider a generalized model which features an amplitude-dependent amplifier gain function. This allows us to determine the total oscillator phase noise in realistic configurations due to noise in both quadratures of the feedback, and to show that it is not necessary to drive the resonator to large oscillation amplitudes in order to eliminate noise in the phase of the feedback.", "doi": "10.1103/PhysRevE.88.062922", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2013-12-23", "series_number": "6", "volume": "88", "issue": "6", "pages": "Art. No. 062922" }, { "id": "authors:9pj48-4fn26", "collection": "authors", "collection_id": "9pj48-4fn26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20131121-112845696", "type": "article", "title": "Coupled motion of microscale and nanoscale elastic objects in a viscous fluid", "author": [ { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Clark", "given_name": "M. T.", "clpid": "Clark-M-T" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study the coupled dynamics of two closely spaced micron or nanoscale elastic objects immersed in a viscous fluid. The dynamics of the elastic objects are coupled through the motion of the surrounding viscous fluid. We consider two cases: (i) one object is driven externally by an imposed harmonic actuation force and the second object is passive and (ii) both objects are driven by a Brownian force to yield stochastic dynamics. Using a harmonic oscillator approximation for the elastic objects and the unsteady Stokes equations to describe the fluid dynamics, we develop analytical expressions for the amplitude and phase of the displacement of the oscillating objects. For the case of an imposed actuation we use an impulse in force to determine the resulting dynamics over all frequencies. For the Brownian-driven objects the stochastic dynamics are found using the fluctuation-dissipation theorem. We validate our theoretical expressions by comparison with results from finite-element numerical simulations of the complete fluid-solid interaction problem. Our results yield interesting features in the amplitude and phase of the displacement of the elastic objects due to the fluid motion. We find that the dynamics depend on the separation of the objects, a measure of the mass loading due to the fluid, and the frequency parameter which acts as a frequency-based Reynolds number. Our results are valid over the range of parameters typical of micron and nanoscale elastic objects in fluid. The range of dynamics found can be understood in terms of the interplay between the viscous and potential components of the fluid flow field described by the unsteady Stokes equation for an oscillating cylinder. For small values of the frequency parameter, typical of nanoscale elastic objects, the dynamics are overdamped due to the dominance of viscous forces over inertial forces. For moderate and large values of the frequency parameter, typical of micron-scale elastic objects, we find that the dynamics of the fluid-coupled objects exhibits an interesting mode splitting to yield a bimodal signature in the amplitude-frequency plots. We find that the mode splitting can be described using a normal mode analysis containing only potential fluid interactions between the cylinders.", "doi": "10.1103/PhysRevE.88.043012", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2013-10-24", "series_number": "4", "volume": "88", "issue": "4", "pages": "Art. No. 043012" }, { "id": "authors:s0hy6-c2q59", "collection": "authors", "collection_id": "s0hy6-c2q59", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130830-094602060", "type": "article", "title": "Quantum-classical transition of correlations of two coupled cavities", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-Tony-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study the difference between quantum and classical behavior in a pair of nonidentical cavities with second-harmonic generation. In the classical limit, each cavity has a limit-cycle solution, in which the photon number oscillates periodically in time. Coupling between the cavities leads to synchronization of the oscillations and classical correlations between the cavities. In the quantum limit, there are quantum correlations due to entanglement. The quantum correlations persist even when the cavities are far off resonance with each other, in stark contrast with the classical case. We also find that the quantum and classical limits are connected by an intermediate regime of almost no correlations. Our results can be extended to a wide variety of quantum models.", "doi": "10.1103/PhysRevA.88.013834", "issn": "1050-2947", "publisher": "American Physical Society", "publication": "Physical Review A", "publication_date": "2013-07-22", "series_number": "1", "volume": "88", "issue": "1", "pages": "Art. No. 013834" }, { "id": "authors:qy1k1-mfb70", "collection": "authors", "collection_id": "qy1k1-mfb70", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130619-111712761", "type": "article", "title": "Frequency precision of two-dimensional lattices of coupled oscillators with spiral patterns", "author": [ { "family_name": "Allen", "given_name": "John-Mark A.", "clpid": "Allen-J-M-A" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Two-dimensional lattices of N synchronized oscillators with reactive coupling are considered as high-precision frequency sources in the case where a spiral pattern is formed. The improvement of the frequency precision is shown to be independent of N for large N, unlike the case of purely dissipative coupling where the improvement is proportional to N, but instead depends on just those oscillators in the core of the spiral that acts as the source region of the waves. Our conclusions are based on numerical simulations of up to N = 29 929 oscillators and analytic results for a continuum approximation to the lattice in an infinite system. We derive an expression for the dependence of the frequency precision on the reactive component of the coupling constant, depending on a single parameter given by fitting the frequency of the spiral waves to the numerical simulations.", "doi": "10.1103/PhysRevE.87.052902", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2013-05-08", "series_number": "5", "volume": "87", "issue": "5", "pages": "Art. No. 052902" }, { "id": "authors:qbe1t-gs417", "collection": "authors", "collection_id": "qbe1t-gs417", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130122-143413954", "type": "article", "title": "Surpassing Fundamental Limits of Oscillators Using Nonlinear Resonators", "author": [ { "family_name": "Villanueva", "given_name": "L. G.", "clpid": "Villanueva-L-G" }, { "family_name": "Kenig", "given_name": "E.", "clpid": "Kenig-E" }, { "family_name": "Karabalin", "given_name": "R. B.", "clpid": "Karabalin-R-B" }, { "family_name": "Matheny", "given_name": "M. H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "In its most basic form an oscillator consists of a resonator driven on resonance, through feedback, to create a periodic signal sustained by a static energy source. The generation of a stable frequency, the basic function of oscillators, is typically achieved by increasing the amplitude of motion of the resonator while remaining within its linear, harmonic regime. Contrary to this conventional paradigm, in this Letter we show that by operating the oscillator at special points in the resonator's anharmonic regime we can overcome fundamental limitations of oscillator performance due to thermodynamic noise as well as practical limitations due to noise from the sustaining circuit. We develop a comprehensive model that accounts for the major contributions to the phase noise of the nonlinear oscillator. Using a nanoelectromechanical system based oscillator, we experimentally verify the existence of a special region in the operational parameter space that enables suppressing the most significant contributions to the oscillator's phase noise, as predicted by our model.", "doi": "10.1103/PhysRevLett.110.177208", "pmcid": "PMC3839326", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2013-04-26", "series_number": "17", "volume": "110", "issue": "17", "pages": "Art. No. 177208" }, { "id": "authors:7s4e8-41418", "collection": "authors", "collection_id": "7s4e8-41418", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121219-133958687", "type": "article", "title": "Optimal operating points of oscillators using nonlinear resonators", "author": [ { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Villanueva", "given_name": "L. G.", "clpid": "Villanueva-L-G" }, { "family_name": "Karabalin", "given_name": "R. B.", "clpid": "Karabalin-R-B" }, { "family_name": "Matheny", "given_name": "M. H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "We demonstrate an analytical method for calculating the phase sensitivity of a class of oscillators whose phase does not affect the time evolution of the other dynamic variables. We show that such oscillators possess the possibility for complete phase noise elimination. We apply the method to a feedback oscillator which employs a high Q weakly nonlinear resonator and provide explicit parameter values for which the feedback phase noise is completely eliminated and others for which there is no amplitude-phase noise conversion. We then establish an operational mode of the oscillator which optimizes its performance by diminishing the feedback noise in both quadratures, thermal noise, and quality factor fluctuations. We also study the spectrum of the oscillator and provide specific results for the case of 1/f noise sources.", "doi": "10.1103/PhysRevE.86.056207", "pmcid": "PMC3839322", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2012-11-13", "series_number": "5", "volume": "86", "issue": "5", "pages": "Art. No. 056207" }, { "id": "authors:88ng0-6bc60", "collection": "authors", "collection_id": "88ng0-6bc60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121101-100531516", "type": "article", "title": "Failure of the free energy relation under a non-Markovian heat bath temperature change", "author": [ { "family_name": "Liang", "given_name": "Cao", "clpid": "Liang-Cao" }, { "family_name": "Cross", "given_name": "Michael", "clpid": "Cross-M-C" }, { "family_name": "Zhi-Gang", "given_name": "Zheng", "clpid": "Zhi-Gang-Zheng" } ], "abstract": "We investigate the free energy relation for a system contacting with a non-Markovian heat bath and find that the validity of the relation sensitively depends on the non-Markovian memory effect, which is especially related to the initial preparation effect. This memory effect drives the statistical distribution of the system out of the initial preparation, even if the system starts from an equilibrium state. This leads to the violation of the free energy relation. A possible way of eliminating this memory effect is proposed.", "doi": "10.1088/1674-1056/21/9/090501", "issn": "1674-1056", "publisher": "IOP Publishing", "publication": "Chinese Physics B", "publication_date": "2012-09", "series_number": "9", "volume": "21", "issue": "9", "pages": "Art. No. 090501" }, { "id": "authors:b0pqk-zy093", "collection": "authors", "collection_id": "b0pqk-zy093", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120723-130614236", "type": "article", "title": "Passive Phase Noise Cancellation Scheme", "author": [ { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Karabalin", "given_name": "R. B.", "clpid": "Karabalin-R-B" }, { "family_name": "Villanueva", "given_name": "L. G.", "clpid": "Villanueva-L-G" }, { "family_name": "Matheny", "given_name": "M. H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "We introduce a new method for reducing phase noise in oscillators, thereby improving their frequency precision. The noise reduction is realized by a passive device consisting of a pair of coupled nonlinear resonating elements that are driven parametrically by the output of a conventional oscillator at a frequency close to the sum of the linear mode frequencies. Above the threshold for parametric instability, the coupled resonators exhibit self-oscillations which arise as a response to the parametric driving, rather than by application of active feedback. We find operating points of the device for which this periodic signal is immune to frequency noise in the driving oscillator, providing a way to clean its phase noise. We present results for the effect of thermal noise to advance a broader understanding of the overall noise sensitivity and the fundamental operating limits.", "doi": "10.1103/PhysRevLett.108.264102", "pmcid": "PMC3839313", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2012-06-29", "series_number": "26", "volume": "108", "issue": "26", "pages": "Art. No. 264102" }, { "id": "authors:afq6d-7q989", "collection": "authors", "collection_id": "afq6d-7q989", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120720-153603430", "type": "article", "title": "Spatiotemporal dynamics of quantum jumps with Rydberg atoms", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-T-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study the nonequilibrium dynamics of quantum jumps in a one-dimensional chain of atoms. Each atom is driven on a strong transition to a short-lived state and on a weak transition to a metastable state. We choose the metastable state to be a Rydberg state so that when an atom jumps to the Rydberg state, it inhibits or enhances jumps in the neighboring atoms. This leads to rich spatiotemporal dynamics that are visible in the fluorescence of the strong transition.", "doi": "10.1103/PhysRevA.85.063822", "issn": "1050-2947", "publisher": "American Physical Society", "publication": "Physical Review A", "publication_date": "2012-06-21", "series_number": "6", "volume": "85", "issue": "6", "pages": "Art. No. 063822" }, { "id": "authors:0p501-jhd13", "collection": "authors", "collection_id": "0p501-jhd13", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120511-101600990", "type": "article", "title": "Improving the frequency precision of oscillators by synchronization", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Improving the frequency precision by synchronizing a lattice of N oscillators with disparate frequencies is studied in the phase reduction limit. In the general case where the coupling is not purely dissipative the synchronized state consists of targetlike waves radiating from a local source, which is a region of higher-frequency oscillators. In this state the improvement of the frequency precision is shown to be independent of N for large N, but instead depends on the disorder and reflects the dependence of the frequency of the synchronized state on just those oscillators in the source region of the waves. These results are obtained by a mapping of the nonlinear phase dynamics onto the linear Anderson problem of the quantum mechanics of electrons on a random lattice in the tight-binding approximation.", "doi": "10.1103/PhysRevE.85.046214", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2012-04-19", "series_number": "4", "volume": "85", "issue": "4", "pages": "Art. No. 046214" }, { "id": "authors:99wnz-1qb20", "collection": "authors", "collection_id": "99wnz-1qb20", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120221-075323895", "type": "monograph", "title": "Pattern formation of quantum jumps with Rydberg atoms", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-T-E" }, { "family_name": "Cross", "given_name": "Michael C.", "clpid": "Cross-M-C" } ], "abstract": "We study the nonequilibrium dynamics of quantum jumps in a one-dimensional chain of atoms. Each atom is driven on a strong transition to a short-lived state and on a weak transition to a metastable state. We choose the metastable state to be a Rydberg state so that when an atom jumps to the Rydberg state, it inhibits or enhances jumps in the neighboring atoms. This leads to rich spatiotemporal dynamics that are visible in the fluorescence of the strong transition. It also allows one to dissipatively prepare Rydberg crystals.", "doi": "10.48550/arXiv.1202.1508", "publisher": "California Institute of Technology", "publication_date": "2012-02-07" }, { "id": "authors:ppj0k-b5q68", "collection": "authors", "collection_id": "ppj0k-b5q68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120302-134815034", "type": "article", "title": "Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles", "author": [ { "family_name": "Li", "given_name": "Dong", "clpid": "Li-Dong" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Zhou", "given_name": "Changsong", "clpid": "Zhou-Changsong" }, { "family_name": "Zheng", "given_name": "Zhigang", "clpid": "Zheng-Zhigang" } ], "abstract": "We investigate two coupled oscillators, each of which shows an attracting heteroclinic cycle in the absence of coupling. The two heteroclinic cycles are nonidentical. Weak coupling can lead to the elimination of the slowing-down state that asymptotically approaches the heteroclinic cycle for a single cycle, giving rise to either quasiperiodic motion with separate frequencies from the two cycles or periodic motion in which the two cycles are synchronized. The synchronization transition, which occurs via a Hopf bifurcation, is not induced by the commensurability of the two cycle frequencies but rather by the disappearance of the weaker frequency oscillation. For even larger coupling the motion changes via a resonant heteroclinic bifurcation to a slowing-down state corresponding to a single attracting heteroclinic orbit. Coexistence of multiple attractors can be found for some parameter regions. These results are of interest in ecological, sociological, neuronal, and other dynamical systems, which have the structure of coupled heteroclinic cycles.", "doi": "10.1103/PhysRevE.85.016215", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2012-01-30", "series_number": "1", "volume": "85", "issue": "1", "pages": "Art. No. 016215" }, { "id": "authors:pxs8f-fsj93", "collection": "authors", "collection_id": "pxs8f-fsj93", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120203-094212729", "type": "article", "title": "Collective Quantum Jumps of Rydberg Atoms", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-T-E" }, { "family_name": "H\u00e4ffner", "given_name": "H.", "clpid": "H\u00e4ffner-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study an open quantum system of atoms with a long-range Rydberg interaction, laser driving, and spontaneous emission. Over time, the system occasionally jumps between a state of low Rydberg population and a state of high Rydberg population. The jumps are inherently collective, and in fact, exist only for a large number of atoms. We explain how entanglement and quantum measurement enable the jumps, which are otherwise classically forbidden.", "doi": "10.1103/PhysRevLett.108.023602", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2012-01-09", "series_number": "2", "volume": "108", "issue": "2", "pages": "Art. No. 023602" }, { "id": "authors:aqk6y-w7c62", "collection": "authors", "collection_id": "aqk6y-w7c62", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111206-084031757", "type": "article", "title": "A Nanoscale Parametric Feedback Oscillator", "author": [ { "family_name": "Villanueva", "given_name": "L. Guillermo", "clpid": "Villanueva-L-G" }, { "family_name": "Karabalin", "given_name": "Rassul B.", "clpid": "Karabalin-R-B" }, { "family_name": "Matheny", "given_name": "Matthew H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Cross", "given_name": "Michael C.", "clpid": "Cross-M-C" }, { "family_name": "Roukes", "given_name": "Michael L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "We describe and demonstrate a new oscillator topology, the parametric feedback oscillator (PFO). The PFO paradigm is applicable to a wide variety of nanoscale devices and opens the possibility of new classes of oscillators employing innovative frequency-determining elements, such as nanoelectromechanical systems (NEMS), facilitating integration with circuitry and system-size reduction. We show that the PFO topology can also improve nanoscale oscillator performance by circumventing detrimental effects that are otherwise imposed by the strong device nonlinearity in this size regime.", "doi": "10.1021/nl2031162", "issn": "1530-6984", "publisher": "American Chemical Society", "publication": "Nano Letters", "publication_date": "2011-11", "series_number": "11", "volume": "11", "issue": "11", "pages": "5054-5059" }, { "id": "authors:vdswt-83138", "collection": "authors", "collection_id": "vdswt-83138", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110926-112200458", "type": "article", "title": "Antiferromagnetic phase transition in a nonequilibrium lattice of Rydberg atoms", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-T-E" }, { "family_name": "H\u00e4ffner", "given_name": "H.", "clpid": "H\u00e4ffner-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study a driven-dissipative system of atoms in the presence of laser excitation to a Rydberg state and spontaneous emission. The atoms interact via the blockade effect, whereby an atom in the Rydberg state shifts the Rydberg level of neighboring atoms. We use mean-field theory to study how the Rydberg population varies in space. As the laser frequency changes, there is a continuous transition between the uniform and antiferromagnetic phases. The nonequilibrium nature also leads to a novel oscillatory phase and bistability between the uniform and antiferromagnetic phases.", "doi": "10.1103/PhysRevA.84.031402", "issn": "1050-2947", "publisher": "American Physical Society", "publication": "Physical Review A", "publication_date": "2011-09-12", "series_number": "3", "volume": "84", "issue": "3", "pages": "Art. No. 031402" }, { "id": "authors:jrygw-jqa95", "collection": "authors", "collection_id": "jrygw-jqa95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110422-113303070", "type": "article", "title": "Pattern Formation with Trapped Ions", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-T-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Ion traps are a versatile tool to study nonequilibrium statistical physics, due to the tunability of dissipation and nonlinearity. We propose an experiment with a chain of ions, where dissipation is provided by laser heating and cooling, while nonlinearity is provided by trap anharmonicity and beam shaping. The dynamics are governed by an equation similar to the complex Ginzburg-Landau equation, except that the reactive nature of the coupling leads to qualitatively different behavior. The system has the unusual feature of being both oscillatory and excitable at the same time. The patterns are observable for realistic experimental parameters despite noise from spontaneous emission. Our scheme also allows controllable experiments with noise and quenched disorder.", "doi": "10.1103/PhysRevLett.106.143001", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2011-04-05", "series_number": "14", "volume": "106", "issue": "14", "pages": "Art. No. 143001" }, { "id": "authors:469j9-m6h59", "collection": "authors", "collection_id": "469j9-m6h59", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20110317-130409894", "type": "article", "title": "Signal Amplification by Sensitive Control of Bifurcation Topology", "author": [ { "family_name": "Karabalin", "given_name": "R. B.", "clpid": "Karabalin-R-B" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Matheny", "given_name": "M. H.", "orcid": "0000-0002-3488-1083", "clpid": "Matheny-M-H" }, { "family_name": "Masmanidis", "given_name": "S. C.", "clpid": "Masmanidis-S-C" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "We describe a novel amplification scheme based on inducing dynamical changes to the topology of a bifurcation diagram of a simple nonlinear dynamical system. We have implemented a first bifurcation-topology amplifier using a coupled pair of parametrically driven high-frequency nanoelectromechanical systems resonators, demonstrating robust small-signal amplification. The principles that underlie bifurcation-topology amplification are simple and generic, suggesting its applicability to a wide variety of physical, chemical, and biological systems.", "doi": "10.1103/PhysRevLett.106.094102", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2011-02-28", "series_number": "9", "volume": "106", "issue": "9", "pages": "Art. No. 094102" }, { "id": "authors:mrbrb-p9g51", "collection": "authors", "collection_id": "mrbrb-p9g51", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100920-105131252", "type": "article", "title": "Vortices and the entrainment transition in the two-dimensional Kuramoto model", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-T-E" }, { "family_name": "Tam", "given_name": "Heywood", "clpid": "Tam-H" }, { "family_name": "Refael", "given_name": "G.", "clpid": "Refael-G" }, { "family_name": "Rogers", "given_name": "Jeffrey L.", "clpid": "Rogers-J-L" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study synchronization in the two-dimensional lattice of coupled phase oscillators with random intrinsic frequencies. When the coupling K is larger than a threshold K_E, there is a macroscopic cluster of frequency-synchronized oscillators. We explain why the macroscopic cluster disappears at K_E. We view the system in terms of vortices, since cluster boundaries are delineated by the motion of these topological defects. In the entrained phase (K>K_E), vortices move in fixed paths around clusters, while in the unentrained phase (K", "doi": "10.1103/PhysRevE.82.036202", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2010-09-02", "series_number": "3", "volume": "82", "issue": "3", "pages": "Art. No. 036202" }, { "id": "authors:az8va-fgj37", "collection": "authors", "collection_id": "az8va-fgj37", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100803-145013910", "type": "article", "title": "Synchronization of oscillators with long-range power law interactions", "author": [ { "family_name": "Chowdhury", "given_name": "Debanjan", "clpid": "Chowdhury-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We present analytical calculations and numerical simulations for the synchronization of oscillators interacting via a long-range power law interaction on a one-dimensional lattice. We have identified the critical value of the power law exponent \u03b1_c across which a transition from a synchronized to an unsynchronized state takes place for a sufficiently strong but finite coupling strength in the large system limit. We find \u03b1_c=3/2. Frequency entrainment and phase ordering are discussed as a function of \u03b1\u22651. The calculations are performed using an expansion about the aligned phase state (spin-wave approximation) and a coarse graining approach. We also generalize the spin-wave results to the d-dimensional problem.", "doi": "10.1103/PhysRevE.82.016205", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2010-07-09", "series_number": "1", "volume": "82", "issue": "1", "pages": "Art. No. 016205" }, { "id": "authors:wvtjp-phv09", "collection": "authors", "collection_id": "wvtjp-phv09", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091124-145326627", "type": "article", "title": "Intrinsic localized modes in parametrically driven arrays of nonlinear resonators", "author": [ { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Malomed", "given_name": "Boris A.", "clpid": "Malomed-B-A" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" } ], "abstract": "We study intrinsic localized modes (ILMs), or solitons, in arrays of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). The analysis is performed using an amplitude equation in the form of a nonlinear Schr\u00f6dinger equation with a term corresponding to nonlinear damping (also known as a forced complex Ginzburg-Landau equation), which is derived directly from the underlying equations of motion of the coupled resonators, using the method of multiple scales. We investigate the creation, stability, and interaction of ILMs, show that they can form bound states, and that under certain conditions one ILM can split into two. Our findings are confirmed by simulations of the underlying equations of motion of the resonators, suggesting possible experimental tests of the theory.", "doi": "10.1103/PhysRevE.80.046202", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2009-10", "series_number": "4", "volume": "80", "issue": "4", "pages": "Art. No. 046202" }, { "id": "authors:nvm0q-c4e20", "collection": "authors", "collection_id": "nvm0q-c4e20", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091124-144207656", "type": "article", "title": "Universality in the one-dimensional chain of phase-coupled oscillators", "author": [ { "family_name": "Lee", "given_name": "Tony E.", "clpid": "Lee-T-E" }, { "family_name": "Refael", "given_name": "G.", "clpid": "Refael-G" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Kogan", "given_name": "Oleg", "clpid": "Kogan-O" }, { "family_name": "Rogers", "given_name": "Jeffrey L.", "clpid": "Rogers-J-L" } ], "abstract": "We apply a recently developed renormalization-group (RG) method to study synchronization in a one-dimensional chain of phase-coupled oscillators in the regime of weak randomness. The RG predicts how oscillators with randomly distributed frequencies and couplings form frequency-synchronized clusters. Although the RG was originally intended for strong randomness, i.e., for distributions with long tails, we find good agreement with numerical simulations even in the regime of weak randomness. We use the RG flow to derive how the correlation length scales with the width of the coupling distribution in the limit of large coupling. This leads to the identification of a universality class of distributions with the same critical exponent v. We also find universal scaling for small coupling. Finally, we show that the RG flow is characterized by a universal approach to the unsynchronized fixed point, which provides physical insight into low-frequency clusters.", "doi": "10.1103/PhysRevE.80.046210", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2009-10", "series_number": "4", "volume": "80", "issue": "4", "pages": "Art. No. 046210" }, { "id": "authors:9stt8-xe437", "collection": "authors", "collection_id": "9stt8-xe437", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091013-093452649", "type": "article", "title": "Synchronization of groups of coupled oscillators with sparse connections", "author": [ { "family_name": "Zheng", "given_name": "Zhigang", "clpid": "Zheng-Zhigang" }, { "family_name": "Feng", "given_name": "Xiaoqin", "clpid": "Feng-Xiaoqin" }, { "family_name": "Ao", "given_name": "Bin", "clpid": "Ao-Bin" }, { "family_name": "Cross", "given_name": "Michael C.", "clpid": "Cross-M-C" } ], "abstract": "Synchronization of groups of coupled oscillators with sparse connections are explored. It is found that different topologies of intergroup couplings may lead to different synchronizability. In the strong-coupling limit, an analytical treatment and criterion is proposed to judge the synchronization between communities of oscillators, and an optimal connection scheme for the group synchronization is given. By varying the intergroup and intragroup coupling strengths, different synchronous phases, i.e., the unsynchronized state, intragroup synchronization, intergroup synchronization, and global synchronization are revealed. The present discussions and results can be applied to study the pattern formation and synchronization of coupled spatiotemporal systems.", "doi": "10.1209/0295-5075/87/50006", "issn": "0295-5075", "publisher": "IOP", "publication": "Europhysics Letters", "publication_date": "2009-09", "series_number": "5", "volume": "87", "issue": "5", "pages": "Art. No. 50006" }, { "id": "authors:e1nq2-0fn39", "collection": "authors", "collection_id": "e1nq2-0fn39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20091020-134924682", "type": "article", "title": "Renormalization group approach to oscillator synchronization", "author": [ { "family_name": "Kogan", "given_name": "Oleg", "clpid": "Kogan-O" }, { "family_name": "Rogers", "given_name": "Jeffrey L.", "clpid": "Rogers-J-L" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Refael", "given_name": "G.", "clpid": "Refael-G" } ], "abstract": "We develop a renormalization group method to investigate synchronization clusters in a one-dimensional chain of nearest-neighbor coupled phase oscillators. The method is best suited for chains with strong disorder in the intrinsic frequencies and coupling strengths. The results are compared with numerical simulations of the chain dynamics and good agreement in several characteristics is found. We apply the renormalization group and simulations to Lorentzian distributions of intrinsic frequencies and couplings and investigate the statistics of the resultant cluster sizes and frequencies, as well as the dependence of the characteristic cluster length upon parameters of these Lorentzian distributions.", "doi": "10.1103/PhysRevE.80.036206", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2009-09", "series_number": "3", "volume": "80", "issue": "3", "pages": "Art. No. 036206" }, { "id": "authors:wr169-h3p93", "collection": "authors", "collection_id": "wr169-h3p93", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090818-082507633", "type": "article", "title": "Nonlinear dynamics and chaos in two coupled nanomechanical resonators", "author": [ { "family_name": "Karabalin", "given_name": "R. B.", "clpid": "Karabalin-R-B" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "Two elastically coupled nanomechanical resonators driven independently near their resonance frequencies show intricate nonlinear dynamics. The dynamics provide a scheme for realizing a nanomechanical system with tunable frequency and nonlinear properties. For large vibration amplitudes, the system develops spontaneous oscillations of amplitude modulation that also show period-doubling transitions and chaos. The complex nonlinear dynamics are quantitatively predicted by a simple theoretical model.", "doi": "10.1103/PhysRevB.79.165309", "issn": "0163-1829", "publisher": "American Physical Society", "publication": "Physical Review B", "publication_date": "2009-04-15", "series_number": "16", "volume": "79", "issue": "16", "pages": "165309" }, { "id": "authors:6vvvm-e5287", "collection": "authors", "collection_id": "6vvvm-e5287", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KENpre09", "type": "article", "title": "Pattern selection in parametrically driven arrays of nonlinear resonators", "author": [ { "family_name": "Kenig", "given_name": "Eyal", "clpid": "Kenig-E" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We study the problem of pattern selection in an array of parametrically driven nonlinear resonators with application to microelectromechanical and nanoelectromechanical systems, using an amplitude equation recently derived by Bromberg, Cross, and Lifshitz [Phys. Rev. E 73, 016214 (2006)]. We describe the transitions between standing-wave patterns of different wave numbers as the drive amplitude is varied either quasistatically, abruptly, or as a linear ramp in time. We find interesting hysteretic effects, which are confirmed by numerical integration of the original equations of motion of the interacting nonlinear resonators.", "doi": "10.1103/PhysRevE.79.026203", "issn": "1539-3755", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2009-02", "series_number": "2", "volume": "79", "issue": "2", "pages": "Art No. 026203" }, { "id": "authors:rbjf2-6np88", "collection": "authors", "collection_id": "rbjf2-6np88", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CARprl07a", "type": "article", "title": "Chaotic Quivering of Micron-Scaled On-Chip Resonators Excited by Centrifugal Optical Pressure", "author": [ { "family_name": "Carmon", "given_name": "Tal", "orcid": "0000-0002-4727-6368", "clpid": "Carmon-T" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Vahala", "given_name": "Kerry J.", "orcid": "0000-0003-1783-1380", "clpid": "Vahala-K-J" } ], "abstract": "Opto-mechanical chaotic oscillation of an on-chip resonator is excited by the radiation-pressure nonlinearity. Continuous optical input, with no external feedback or modulation, excites chaotic vibrations in very different geometries of the cavity (both tori and spheres) and shows that opto-mechanical chaotic oscillations are an intrinsic property of optical microcavities. Measured phenomena include period doubling, a spectral continuum, aperiodic oscillations, and complex trajectories. The rate of exponential divergence from a perturbed initial condition (Lyapunov exponent) is calculated. Continuous improvements in cavities mean that such chaotic oscillations can be expected in the future with many other platforms, geometries, and frequency spans.", "doi": "10.1103/PhysRevLett.98.167203", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2007-04-20", "series_number": "16", "volume": "98", "issue": "16", "pages": "Art. No. 167203" }, { "id": "authors:smpam-p8493", "collection": "authors", "collection_id": "smpam-p8493", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PAUpre07a", "type": "article", "title": "Extensive chaos in Rayleigh-B\u00e9nard convection", "author": [ { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Einarsson", "given_name": "M. I.", "clpid": "Einarsson-M-I" }, { "family_name": "Fischer", "given_name": "P. F.", "clpid": "Fischer-P-F" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Using large-scale numerical calculations we explore spatiotemporal chaos in Rayleigh-B\u00e9nard convection for experimentally relevant conditions. We calculate the spectrum of Lyapunov exponents and the Lyapunov dimension describing the chaotic dynamics of the convective fluid layer at constant thermal driving over a range of finite system sizes. Our results reveal that the dynamics of fluid convection is truly chaotic for experimental conditions as illustrated by a positive leading-order Lyapunov exponent. We also find the chaos to be extensive over the range of finite-sized systems investigated as indicated by a linear scaling between the Lyapunov dimension of the chaotic attractor and the system size.", "doi": "10.1103/PhysRevE.75.045203", "issn": "1539-3755", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2007-04-01", "series_number": "4", "volume": "75", "issue": "4", "pages": "Art. No. 045203(R)" }, { "id": "authors:67sen-t0f64", "collection": "authors", "collection_id": "67sen-t0f64", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100819-131952947", "type": "book_section", "title": "BioNEMS: Nanomechanical systems for single-molecule biophysics", "book_title": "Controlled nanoscale motion Nobel Symposium 131", "author": [ { "family_name": "Arlett", "given_name": "J. L.", "clpid": "Arlett-J-L" }, { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Solomon", "given_name": "J. E.", "clpid": "Solomon-J-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Fraser", "given_name": "S. E.", "orcid": "0000-0002-5377-0223", "clpid": "Fraser-S-E" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "contributor": [ { "family_name": "Linke", "given_name": "H.", "clpid": "Linke-H" }, { "family_name": "M\u00e5nsson", "given_name": "Alf", "clpid": "M\u00e5nsson-A" } ], "abstract": "Techniques from nanoscience now enable the creation of ultrasmall electronic\ndevices. Among these, nanoelectromechanical systems (NEMS) in particular\noffer unprecedented opportunities for sensitive chemical, biological, and physical\nmeasurements [1]. For vacuum-based applications NEMS provide extremely high\nforce and mass sensitivity, ultimately below the attonewton and single-Dalton level\nrespectively. In fluidic media, even though the high quality factors attainable in\nvacuum become precipitously damped due to fluid coupling, extremely small device\nsize and high compliance still yield force sensitivity at the piconewton level -\ni.e., smaller than that, on average, required to break individual hydrogen bonds that\nare the fundamental structural elements underlying molecular recognition processes.\nA profound and unique new feature of nanoscale fluid-based mechanical sensors is\nthat they offer the advantage of unprecedented signal bandwidth (\u00bb1 MHz), even\nat piconewton force levels. Their combined sensitivity and temporal resolution is\ndestined to enable real-time observations of stochastic single-molecular biochemical\nprocesses down to the sub-microsecond regime [2].", "doi": "10.1007/3-540-49522-3_12", "isbn": "978-3-540-49521-5", "publisher": "Berlin", "place_of_publication": "New York", "publication_date": "2007", "pages": "241-270" }, { "id": "authors:r3aw6-e1825", "collection": "authors", "collection_id": "r3aw6-e1825", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SCHEpre06", "type": "article", "title": "Lyapunov exponents for small aspect ratio Rayleigh-B\u00e9nard convection", "author": [ { "family_name": "Scheel", "given_name": "J. D.", "clpid": "Scheel-J-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Leading order Lyapunov exponents and their corresponding eigenvectors have been computed numerically for small aspect ratio, three-dimensional Rayleigh-Benard convection cells with no-slip boundary conditions. The parameters are the same as those used by Ahlers and Behringer [Phys. Rev. Lett. 40, 712 (1978)] and Gollub and Benson [J. Fluid Mech. 100, 449 (1980)] in their work on a periodic time dependence in Rayleigh-Benard convection cells. Our work confirms that the dynamics in these cells truly are chaotic as defined by a positive Lyapunov exponent. The time evolution of the leading order Lyapunov eigenvector in the chaotic regime will also be discussed. In addition we study the contributions to the leading order Lyapunov exponent for both time periodic and aperiodic states and find that while repeated dynamical events such as dislocation creation/annihilation and roll compression do contribute to the short time Lyapunov exponent dynamics, they do not contribute to the long time Lyapunov exponent. We find instead that nonrepeated events provide the most significant contribution to the long time leading order Lyapunov exponent.", "doi": "10.1103/PhysRevE.74.066301", "issn": "1539-3755", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2006-12-01", "series_number": "6", "volume": "74", "issue": "6", "pages": "Art. No. 066301" }, { "id": "authors:5hnjf-9yj38", "collection": "authors", "collection_id": "5hnjf-9yj38", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191009-154950074", "type": "article", "title": "Synchronization on coupled dynamical networks", "author": [ { "family_name": "Zheng", "given_name": "Zhi-gang", "clpid": "Zheng-Zhi-gang" }, { "family_name": "Feng", "given_name": "Xiao-qin", "clpid": "Feng-Xiao-qin" }, { "family_name": "Ao", "given_name": "Bin", "clpid": "Ao-Bin" }, { "family_name": "Cross", "given_name": "Michael C.", "clpid": "Cross-M-C" } ], "abstract": "In this paper, partial synchronization (PaS) in networks of coupled chaotic oscillator systems and synchronization in sparsely coupled spatiotemporal systems are explored. For the PaS, we reveal that the existence of PaS patterns depends on the symmetry property of the network topology, while the emergence of the PaS pattern depends crucially on the stability of the corresponding solution. An analytical criterion in judging the stability of PaS state on a given network are proposed in terms of a comparison between the Lyapunov exponent spectrum of the PaS manifold and that of the transversal manifold. The competition and selections of the PaS patterns induced by the presence of multiple topological symmetries of the network are studied in terms of the criterion. The phase diagram in distinguishing the synchronous and the asynchronous states is given. The criterion in judging PaS is further applied to the study of synchronization of two sparsely coupled spatiotemporal chaotic systems. Different synchronization regimes are distinguished. The present study reveals the intrinsic collective bifurcation of coupled dynamical systems prior to the emergence of global synchronization.", "doi": "10.1007/s11467-006-0047-3", "issn": "1673-3487", "publisher": "Springer", "publication": "Frontiers of Physics in China", "publication_date": "2006-12", "series_number": "4", "volume": "1", "issue": "4", "pages": "458-467" }, { "id": "authors:ee7fm-sx467", "collection": "authors", "collection_id": "ee7fm-sx467", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CARomems06a", "type": "book_section", "title": "Opto-Mechanical Chaotic Behaviour of Micron-Scaled On-Chip Resonators", "book_title": "IEEE/LEOS International Conference on Optical MEMS and Their Applications Conference, 2006 (OMEMS '06)", "author": [ { "family_name": "Carmon", "given_name": "Tal", "orcid": "0000-0002-4727-6368", "clpid": "Carmon-T" }, { "family_name": "Cross", "given_name": "Michael C.", "clpid": "Cross-M-C" }, { "family_name": "Vahala", "given_name": "Kerry J.", "orcid": "0000-0003-1783-1380", "clpid": "Vahala-K-J" } ], "abstract": "Opto-mechanical vibration of an on-chip oscillator is experimentally excited by radiation-pressure nonlinearity to a regime where oscillation is chaotic. Period-doubling and broad power spectra are measured in spherical and toroidal-resonators.", "doi": "10.1109/OMEMS.2006.1708262", "isbn": "0-7803-9562-X", "publisher": "IEEE", "place_of_publication": "Piscataway, NJ", "publication_date": "2006-10-09", "pages": "56-57" }, { "id": "authors:b5gx3-p7p56", "collection": "authors", "collection_id": "b5gx3-p7p56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PAUnano06", "type": "article", "title": "The stochastic dynamics of micron and nanoscale elastic cantilevers in fluid: fluctuations from dissipation", "author": [ { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Clark", "given_name": "M. T.", "clpid": "Clark-M-T" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The stochastic dynamics of micron and nanoscale cantilevers immersed in a viscous fluid are quantified. Analytical results are presented for long slender cantilevers driven by Brownian noise. The spectral density of the noise force is not assumed to be white and the frequency dependence of the noise force is determined from the fluctuation-dissipation theorem. The analytical results are shown to be useful for the micron scale cantilevers that are commonly used in atomic force microscopy. A general thermodynamic approach is developed that is valid for cantilevers of arbitrary geometry as well as for arrays of multiple cantilevers whose stochastic motion is coupled through the fluid. It is shown that the fluctuation-dissipation theorem permits the calculation of stochastic quantities via straightforward deterministic methods. The thermodynamic approach is used with deterministic finite element numerical simulations to quantify the auto-correlation and noise spectrum of cantilever fluctuations for a single micron scale cantilever and the cross-correlations and noise spectra of fluctuations for an array of two experimentally motivated nanoscale cantilevers as a function of cantilever separation. The results are used to quantify the noise reduction possible using correlated measurements with two closely spaced nanoscale cantilevers.", "doi": "10.1088/0957-4484/17/17/037", "issn": "0957-4484", "publisher": "IOP", "publication": "Nanotechnology", "publication_date": "2006-09-14", "series_number": "17", "volume": "17", "issue": "17", "pages": "4502-4513" }, { "id": "authors:nk96y-1da06", "collection": "authors", "collection_id": "nk96y-1da06", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:URBprb06", "type": "article", "title": "Perturbation of magnetostatic modes observed by ferromagnetic resonance force microscopy", "author": [ { "family_name": "Urban", "given_name": "R.", "clpid": "Urban-R" }, { "family_name": "Putilin", "given_name": "A.", "clpid": "Putilin-A" }, { "family_name": "Wigen", "given_name": "P. E.", "clpid": "Wigen-P-E" }, { "family_name": "Liou", "given_name": "S.-H.", "clpid": "Liou-S-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Hammel", "given_name": "P.C.", "clpid": "Hammel-P-C" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "Magnetostatic modes of yttrium iron garnet (YIG) films are investigated by ferromagnetic resonance force microscopy. A thin-film \"probe\" magnet at the tip of a compliant cantilever introduces a local inhomogeneity in the internal field of the YIG sample. This influences the shape of the sample's magnetostatic modes, thereby measurably perturbing the strength of the force coupled to the cantilever. We present a theoretical model that explains these observations; it shows that the tip-induced variation of the internal field creates either a local \"potential barrier\" or \"potential well\" for the magnetostatic waves. The data and model together indicate that local magnetic imaging of ferromagnets is possible, even in the presence of long-range spin coupling, through the introduction of localized magnetostatic modes predicted to arise from sufficiently strong tip fields.", "doi": "10.1103/PhysRevB.73.212410", "issn": "1098-0121", "publisher": "American Physical Society", "publication": "Physical Review B", "publication_date": "2006-06-01", "series_number": "21", "volume": "73", "issue": "21", "pages": "Art. No. 212410" }, { "id": "authors:24dbt-pdq51", "collection": "authors", "collection_id": "24dbt-pdq51", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BECpre06", "type": "article", "title": "Effect of the centrifugal force on domain chaos in Rayleigh-B\u00e9nard convection", "author": [ { "family_name": "Becker", "given_name": "Nathan", "clpid": "Becker-N" }, { "family_name": "Scheel", "given_name": "J. D.", "clpid": "Scheel-J-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Ahlers", "given_name": "Guenter", "clpid": "Ahlers-G" } ], "abstract": "Experiments and simulations from a variety of sample sizes indicated that the centrifugal force significantly affects the domain-chaos state observed in rotating Rayleigh-B\u00e9nard convection-patterns. In a large-aspect-ratio sample, we observed a hybrid state consisting of domain chaos close to the sample center, surrounded by an annulus of nearly stationary nearly radial rolls populated by occasional defects reminiscent of undulation chaos. Although the Coriolis force is responsible for domain chaos, by comparing experiment and simulation we show that the centrifugal force is responsible for the radial rolls. Furthermore, simulations of the Boussinesq equations for smaller aspect ratios neglecting the centrifugal force yielded a domain precession-frequency f~epsilon\u00b5 with \u00b5~=1 as predicted by the amplitude-equation model for domain chaos, but contradicted by previous experiment. Additionally the simulations gave a domain size that was larger than in the experiment. When the centrifugal force was included in the simulation, \u00b5 and the domain size were consistent with experiment.", "doi": "10.1103/PhysRevE.73.066309", "issn": "1539-3755", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2006-06-01", "series_number": "6", "volume": "73", "issue": "6", "pages": "Art. No. 066309" }, { "id": "authors:ptphv-zpv79", "collection": "authors", "collection_id": "ptphv-zpv79", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpre06", "type": "article", "title": "Synchronization by reactive coupling and nonlinear frequency pulling", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Rogers", "given_name": "J. L.", "clpid": "Rogers-J-L" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Zumdieck", "given_name": "A.", "clpid": "Zumdieck-A" } ], "abstract": "We present a detailed analysis of a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling. We study the model for the mean field case of all-to-all coupling, deriving results for the initial onset of synchronization as the coupling or nonlinearity increase, and conditions for the existence of the completely synchronized state when all the oscillators evolve with the same frequency. Explicit results are derived for the Lorentzian, triangular, and top-hat distributions of oscillator frequencies. Numerical simulations are used to construct complete phase diagrams for these distributions.", "doi": "10.1103/PhysRevE.73.036205", "issn": "1539-3755", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2006-03-01", "series_number": "3", "volume": "73", "issue": "3", "pages": "Art. No. 036205" }, { "id": "authors:mc4hm-39764", "collection": "authors", "collection_id": "mc4hm-39764", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BROpre06", "type": "article", "title": "Response of discrete nonlinear systems with many degrees of freedom", "author": [ { "family_name": "Bromberg", "given_name": "Yaron", "clpid": "Bromberg-Y" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" } ], "abstract": "We study the response of a large array of coupled nonlinear oscillators to parametric excitation, motivated by the growing interest in the nonlinear dynamics of microelectromechanical and nanoelectromechanical systems (MEMS and NEMS). Using a multiscale analysis, we derive an amplitude equation that captures the slow dynamics of the coupled oscillators just above the onset of parametric oscillations. The amplitude equation that we derive here from first principles exhibits a wave-number dependent bifurcation similar in character to the behavior known to exist in fluids undergoing the Faraday wave instability. We confirm this behavior numerically and make suggestions for testing it experimentally with MEMS and NEMS resonators.", "doi": "10.1103/PhysRevE.73.016214", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2006-01-01", "series_number": "1", "volume": "73", "issue": "1", "pages": "Art. No. 016214" }, { "id": "authors:s8xcb-q3646", "collection": "authors", "collection_id": "s8xcb-q3646", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SCHEpre05", "type": "article", "title": "Scaling laws for rotating Rayleigh-B\u00e9nard convection", "author": [ { "family_name": "Scheel", "given_name": "J. D.", "clpid": "Scheel-J-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Numerical simulations of large aspect ratio, three-dimensional rotating Rayleigh-B\u00e9nard convection for no-slip boundary conditions have been performed in both cylinders and periodic boxes. We have focused near the threshold for the supercritical bifurcation from the conducting state to a convecting state exhibiting domain chaos. A detailed analysis of these simulations has been carried out and is compared with experimental results, as well as predictions from multiple scale perturbation theory. We find that the time scaling law agrees with the theoretical prediction, which is in contradiction to experimental results. We also have looked at the scaling of defect lengths and defect glide velocities. We find a separation of scales in defect diameters perpendicular and parallel to the rolls as expected, but the scaling laws for the two different lengths are in contradiction to theory. The defect velocity scaling law agrees with our theoretical prediction from multiple scale perturbation theory.", "doi": "10.1103/PhysRevE.72.056315", "issn": "1539-3755", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2005-11-01", "series_number": "5", "volume": "72", "issue": "5", "pages": "Art. No. 056315" }, { "id": "authors:apkqh-3xx70", "collection": "authors", "collection_id": "apkqh-3xx70", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CHIpre05", "type": "article", "title": "Enhanced tracer transport by the spiral defect chaos state of a convecting fluid", "author": [ { "family_name": "Chiam", "given_name": "K.-H.", "clpid": "Chiam-K-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Greenside", "given_name": "H. S.", "clpid": "Greenside-H-S" }, { "family_name": "Fischer", "given_name": "P. F.", "clpid": "Fischer-P-F" } ], "abstract": "To understand how spatiotemporal chaos may modify material transport, we use direct numerical simulations of the three-dimensional Boussinesq equations and of an advection-diffusion equation to study the transport of a passive tracer by the spiral defect chaos state of a convecting fluid. The simulations show that the transport is diffusive and is enhanced by the spatiotemporal chaos. The enhancement in tracer diffusivity follows two regimes. For large P\u00e9clet numbers (that is, small molecular diffusivities of the tracer), we find that the enhancement is proportional to the P\u00e9clet number. For small P\u00e9clet numbers, the enhancement is proportional to the square root of the P\u00e9clet number. We explain the presence of these two regimes in terms of how the local transport depends on the local wave numbers of the convection rolls. For large P\u00e9clet numbers, we further find that defects cause the tracer diffusivity to be enhanced locally in the direction orthogonal to the local wave vector but suppressed in the direction of the local wave vector.", "doi": "10.1103/PhysRevE.71.036205", "issn": "1539-3755", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2005-03-01", "series_number": "3", "volume": "71", "issue": "3", "pages": "Art. No. 036205" }, { "id": "authors:sw0kd-cmv95", "collection": "authors", "collection_id": "sw0kd-cmv95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprl04", "type": "article", "title": "Synchronization by Nonlinear Frequency Pulling", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Zumdieck", "given_name": "A.", "clpid": "Zumdieck-A" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Rogers", "given_name": "J. L.", "clpid": "Rogers-J-L" } ], "abstract": "We analyze a model for the synchronization of nonlinear oscillators due to reactive coupling and nonlinear frequency pulling motivated by the physics of arrays of nanoscale oscillators. We study the model for the mean field case of all-to-all coupling, deriving results for the onset of synchronization as the coupling or nonlinearity increase, and the fully locked state when all the oscillators evolve with the same frequency.", "doi": "10.1103/PhysRevLett.93.224101", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2004-11-26", "series_number": "22", "volume": "93", "issue": "22", "pages": "Art. No. 224101" }, { "id": "authors:jnsdm-he167", "collection": "authors", "collection_id": "jnsdm-he167", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SANprb04", "type": "article", "title": "Quantum nondemolition measurement of Fock states of mesoscopic mechanical oscillators", "author": [ { "family_name": "Santamore", "given_name": "D. H.", "clpid": "Santamore-D-H" }, { "family_name": "Doherty", "given_name": "A. C.", "clpid": "Doherty-A-C" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We investigate a scheme that makes a quantum nondemolition (QND) measurement of the excitation level of a mesoscopic mechanical oscillator by utilizing the anharmonic coupling between two beam bending modes. The nonlinear coupling between the two modes shifts the resonant frequency of the readout oscillator in proportion to the excitation level of the system oscillator. This frequency shift may be detected as a phase shift of the readout oscillation when driven on resonance. We derive an equation for the reduced density matrix of the system oscillator, and use this to study the conditions under which discrete jumps in the excitation level occur. The appearance of jumps in the actual quantity measured is also studied using the method of quantum trajectories. We consider the feasibility of the scheme for experimentally accessible parameters.", "doi": "10.1103/PhysRevB.70.144301", "issn": "1098-0121", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "2004-10-01", "series_number": "14", "volume": "70", "issue": "14", "pages": "Art. No. 144301" }, { "id": "authors:241gd-t6n90", "collection": "authors", "collection_id": "241gd-t6n90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PAUprl04a", "type": "article", "title": "Stochastic dynamics of nanoscale mechanical oscillators immersed in a viscous fluid", "author": [ { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The stochastic response of nanoscale oscillators of arbitrary geometry immersed in a viscous fluid is studied. Using the fluctuation-dissipation theorem, it is shown that deterministic calculations of the governing fluid and solid equations can be used in a straightforward manner to directly calculate the stochastic response that would be measured in experiment. We use this approach to investigate the fluid coupled motion of single and multiple cantilevers with experimentally motivated geometries.", "doi": "10.1103/PhysRevLett.92.235501", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2004-06-11", "series_number": "23", "volume": "92", "issue": "23", "pages": "Art. No. 235501" }, { "id": "authors:c1k8f-k2w52", "collection": "authors", "collection_id": "c1k8f-k2w52", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SCHEpre03", "type": "article", "title": "Traveling waves in rotating Rayleigh-B\u00e9nard convection: Analysis of modes and mean flow", "author": [ { "family_name": "Scheel", "given_name": "J. D.", "clpid": "Scheel-J-D" }, { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Fischer", "given_name": "P. F.", "clpid": "Fischer-P-F" } ], "abstract": "Numerical simulations of the Boussinesq equations with rotation for realistic no-slip boundary conditions and a finite annular domain are presented. These simulations reproduce traveling waves observed experimentally. Traveling waves are studied near threshhold by using the complex Ginzburg-Landau equation (CGLE): a mode analysis enables the CGLE coefficients to be determined. The CGLE coefficients are compared with previous experimental and theoretical results. Mean flows are also computed and found to be more significant as the Prandtl number decreases (from sigma=6.4 to sigma=1). In addition, the mean flow around the outer radius of the annulus appears to be correlated with the mean flow around the inner radius.", "doi": "10.1103/PhysRevE.68.066216", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2003-12-01", "series_number": "6", "volume": "68", "issue": "6", "pages": "Art. No. 066216" }, { "id": "authors:3585y-xqy07", "collection": "authors", "collection_id": "3585y-xqy07", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CHIpre03", "type": "article", "title": "Mean flow and spiral defect chaos in Rayleigh-B\u00e9nard convection", "author": [ { "family_name": "Chiam", "given_name": "K.-H.", "clpid": "Chiam-K-H" }, { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Greenside", "given_name": "H. S.", "clpid": "Greenside-H-S" } ], "abstract": "We describe a numerical procedure to construct a modified velocity field that does not have any mean flow. Using this procedure, we present two results. First, we show that, in the absence of the mean flow, spiral defect chaos collapses to a stationary pattern comprising textures of stripes with angular bends. The quenched patterns are characterized by mean wave numbers that approach those uniquely selected by focus-type singularities, which, in the absence of the mean flow, lie at the zigzag instability boundary. The quenched patterns also have larger correlation lengths and are comprised of rolls with less curvature. Secondly, we describe how the mean flow can contribute to the commonly observed phenomenon of rolls terminating perpendicularly into lateral walls. We show that, in the absence of the mean flow, rolls begin to terminate into lateral walls at an oblique angle. This obliqueness increases with the Rayleigh number.", "doi": "10.1103/PhysRevE.67.056206", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2003-05-01", "series_number": "5", "volume": "67", "issue": "5", "pages": "Art. No. 056206" }, { "id": "authors:c2h5k-80z42", "collection": "authors", "collection_id": "c2h5k-80z42", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:LIFprb03", "type": "article", "title": "Response of parametrically driven nonlinear coupled oscillators with application to micromechanical and nanomechanical resonator arrays", "author": [ { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The response of a coupled array of nonlinear oscillators to parametric excitation is calculated in the weak nonlinear limit using secular perturbation theory. Exact results for small arrays of oscillators are used to guide the analysis of the numerical integration of the model equations of motion for large arrays. The results provide a qualitative explanation for a recent experiment [Buks and Roukes, J. Microelectromech. Syst. 11, 802 (2002)] involving a parametrically excited micromechanical resonator array. Future experiments are suggested which could provide quantitative tests of the theoretical predictions.", "doi": "10.1103/PhysRevB.67.134302", "issn": "1098-0121", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "2003-04-01", "series_number": "13", "volume": "67", "issue": "13", "pages": "Art. No. 134302" }, { "id": "authors:bhm7z-2fj83", "collection": "authors", "collection_id": "bhm7z-2fj83", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:ZHEprl02", "type": "article", "title": "Collective directed transport of symmetrically coupled lattices in symmetric periodic potentials", "author": [ { "family_name": "Zheng", "given_name": "Zhigang", "clpid": "Zheng-Zhigang" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Hu", "given_name": "Gang", "clpid": "Hu-Gang" } ], "abstract": "A mechanism responsible for the directed transport of symmetrically coupled lattices in a symmetric periodic potential is proposed. Under the action of an external wave that breaks the spatiotemporal symmetry and introduces inhomogeneities of the lattice, a net unidirectional current can be observed, originating from a collaboration of the lattice (periodic potential and mutual interactions) and the wave (amplitude, frequency, and phase shifts). Mode-locking induced resonant steps are observed and predicted. The directed transport can be optimized and furthermore controlled by suitably adjusting the parameters of the system.", "doi": "10.1103/PhysRevLett.89.154102", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2002-10-07", "series_number": "15", "volume": "89", "issue": "15", "pages": "Art. No. 154102" }, { "id": "authors:j982b-f8h68", "collection": "authors", "collection_id": "j982b-f8h68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PAUpre02", "type": "article", "title": "Rayleigh-B\u00e9nard convection with a radial ramp in plate separation", "author": [ { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Fischer", "given_name": "P. F.", "clpid": "Fischer-P-F" } ], "abstract": "Pattern formation in Rayleigh-B\u00e9nard convection in a large-aspect-ratio cylinder with a radial ramp in the plate separation is studied analytically and numerically by performing numerical simulations of the Boussinesq equations. A horizontal mean flow and a vertical large scale counterflow are quantified and used to understand the pattern wave number. Our results suggest that the mean flow, generated by amplitude gradients, plays an important role in the roll compression observed as the control parameter is increased. Near threshold, the mean flow has a quadrupole dependence with a single vortex in each quadrant while away from threshold the mean flow exhibits an octupole dependence with a counterrotating pair of vortices in each quadrant. This is confirmed analytically using the amplitude equation and Cross-Newell mean flow equation. By performing numerical experiments, the large scale counterflow is also found to aid in the roll compression away from threshold but to a much lesser degree. Our results yield an understanding of the pattern wave numbers observed in experiment away from threshold and suggest that near threshold the mean flow and large scale counterflow are not responsible for the observed shift to smaller than critical wave numbers.", "doi": "10.1103/PhysRevE.66.046210", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2002-10-01", "series_number": "4", "volume": "66", "issue": "4", "pages": "Art. No. 046210" }, { "id": "authors:wvzp2-js109", "collection": "authors", "collection_id": "wvzp2-js109", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SANprb02", "type": "article", "title": "Surface scattering analysis of phonon transport in the quantum limit using an elastic model", "author": [ { "family_name": "Santamore", "given_name": "D. H.", "clpid": "Santamore-D-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We have investigated the effect on phonon energy transport in mesoscopic systems and the reduction in the thermal conductance in the quantum limit due to phonon scattering by surface roughness, using full three-dimensional elasticity theory for an elastic beam with a rectangular cross section. At low frequencies, we find power laws for the scattering coefficients that are strongly mode dependent, and different from the \u03c92 dependence, deriving from Rayleigh scattering of scalar waves, that is often assumed. The scattering gives contributions to the reduction in thermal conductance with the same power laws. At higher frequencies, the scattering coefficients becomes large at the onset frequency of each mode due to the flat dispersion here. We use our results to attempt a quantitative understanding of the suppression of the thermal conductance from the universal value observed in the experiment.", "doi": "10.1103/PhysRevB.66.144302", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "2002-10-01", "series_number": "14", "volume": "66", "issue": "14", "pages": "Art. No. 144302" }, { "id": "authors:kfvpt-3kj83", "collection": "authors", "collection_id": "kfvpt-3kj83", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:ZHEpre02", "type": "article", "title": "Transitions from partial to complete generalized synchronizations in bidirectionally coupled chaotic oscillators", "author": [ { "family_name": "Zheng", "given_name": "Zhigang", "clpid": "Zheng-Zhigang" }, { "family_name": "Wang", "given_name": "Xingang", "clpid": "Wang-Xingang" }, { "family_name": "Cross", "given_name": "Michael C.", "clpid": "Cross-M-C" } ], "abstract": "Generalized synchronization in an array of mutually (bidirectionally) coupled nonidentical chaotic oscillators is studied. Coupled Lorenz oscillators and coupled Lorenz-Rossler oscillators are adopted as our working models. With increasing the coupling strengths, the system experiences a cascade of transitions from the partial to the global generalized synchronizations, i.e., different oscillators are gradually entrained through a clustering process. This scenario of transitions reveals an intrinsic self-organized order in groups of interacting units, which generalizes the idea of generalized synchronizations in drive-response systems.", "doi": "10.1103/PhysRevE.65.056211", "issn": "1063-651X", "publisher": "American Physical Society", "publication": "Physical Review E", "publication_date": "2002-05", "series_number": "6", "volume": "65", "issue": "6", "pages": "Art. No. 056211" }, { "id": "authors:90f4q-ax064", "collection": "authors", "collection_id": "90f4q-ax064", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PURpre01", "type": "article", "title": "Nonequilibrium dynamics in the complex Ginzburg-Landau equation", "author": [ { "family_name": "Puri", "given_name": "Sanjay", "clpid": "Puri-S" }, { "family_name": "Das", "given_name": "Subir K.", "clpid": "Das-S-K" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Results from a comprehensive analytical and numerical study of nonequilibrium dynamics in the two-dimensional complex Ginzburg-Landau equation have been presented. In particular, spiral defects have been used to characterize the domain growth law and the evolution morphology. An asymptotic analysis of the single-spiral correlation function shows a sequence of singularities\u2014analogous to those seen for time-dependent Ginzburg-Landau models with O(n) symmetry, where n is even.", "doi": "10.1103/PhysRevE.64.056140", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2001-11-01", "series_number": "5", "volume": "64", "issue": "5", "pages": "Art. No. 056140" }, { "id": "authors:wt9hk-0gb78", "collection": "authors", "collection_id": "wt9hk-0gb78", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:PAUprl01", "type": "article", "title": "Power-law behavior of power spectra in low Prandtl number Rayleigh-B\u00e9nard convection", "author": [ { "family_name": "Paul", "given_name": "M. R.", "clpid": "Paul-M-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Fischer", "given_name": "P. F.", "clpid": "Fischer-P-F" }, { "family_name": "Greenside", "given_name": "H. S.", "clpid": "Greenside-H-S" } ], "abstract": "The origin of the power-law decay measured in the power spectra of low Prandtl number Rayleigh-Benard convection near the onset of chaos is addressed using long time numerical simulations of the three-dimensional Boussinesq equations in cylindrical domains. The power law is found to arise from quasidiscontinuous changes in the slope of the time series of the heat transport associated with the nucleation of dislocation pairs and roll pinch-off events. For larger frequencies, the power spectra decay exponentially as expected for time continuous deterministic dynamics.", "doi": "10.1103/PhysRevLett.87.154501", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2001-10-08", "series_number": "15", "volume": "87", "issue": "15", "pages": "Art. No. 154501" }, { "id": "authors:8mpgd-2zx03", "collection": "authors", "collection_id": "8mpgd-2zx03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:DASpre01", "type": "article", "title": "Nonequilibrium dynamics of the complex Ginzburg-Landau equation: Analytical results", "author": [ { "family_name": "Das", "given_name": "Subir K.", "clpid": "Das-S-K" }, { "family_name": "Puri", "given_name": "Sanjay", "clpid": "Puri-S" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We present a detailed analytical and numerical study of nonequilibrium dynamics for the complex Ginzburg-Landau equation. In particular, we characterize evolution morphologies using spiral defects. This paper is the first in a two-stage exposition. Here, we present analytical results for the correlation function arising from a single-spiral morphology. We also critically examine the utility of the Gaussian auxiliary field ansatz in characterizing a multispiral morphology. In the next paper of this exposition we will present detailed numerical results.", "doi": "10.1103/PhysRevE.64.046206", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2001-10-01", "series_number": "4", "volume": "64", "issue": "4", "pages": "Art. No. 046206" }, { "id": "authors:ce6cf-0w736", "collection": "authors", "collection_id": "ce6cf-0w736", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SANprl01", "type": "article", "title": "Effect of phonon scattering by surface roughness on the universal thermal conductance", "author": [ { "family_name": "Santamore", "given_name": "D. H.", "clpid": "Santamore-D-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The effect of phonon scattering by surface roughness on the thermal conductance in mesoscopic systems at low temperatures is calculated using full elasticity theory. The low frequency behavior of the scattering shows novel power law dependences arising from the unusual properties of the elastic modes. This leads to new predictions for the low temperature depression of the thermal conductance below the ideal universal value. Comparison with the data of Schwab et al. [Nature (London) 404, 974 (2000)] suggests that surface roughness on a scale of the width of the thermal pathway is important in the experiment.", "doi": "10.1103/PhysRevLett.87.115502", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "2001-09-10", "series_number": "11", "volume": "87", "issue": "11", "pages": "Art. No. 115502" }, { "id": "authors:4jzyv-nhr72", "collection": "authors", "collection_id": "4jzyv-nhr72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprb01", "type": "article", "title": "Elastic wave transmission at an abrupt junction in a thin plate with application to heat transport and vibrations in mesoscopic systems", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" } ], "abstract": "The transmission coefficient for vibrational waves crossing an abrupt junction between two thin elastic plates of different widths is calculated. These calculations are relevant to ballistic phonon thermal transport at low temperatures in mesoscopic systems and the Q for vibrations in mesoscopic oscillators. Complete results are derived in a simple scalar model of the elastic waves, and results for long-wavelength modes are obtained using full elasticity theory. We suggest that thin-plate elasticity theory provides a useful and tractable approximation to the three-dimensional geometry.", "doi": "10.1103/PhysRevB.64.085324", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "2001-08-15", "series_number": "8", "volume": "64", "issue": "8", "pages": "Art. No. 085324" }, { "id": "authors:ad4vh-7py44", "collection": "authors", "collection_id": "ad4vh-7py44", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SANprb01", "type": "article", "title": "Effect of surface roughness on the universal thermal conductance", "author": [ { "family_name": "Santamore", "given_name": "D. H.", "clpid": "Santamore-D-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We explain the reduction of the thermal conductance below the predicted universal value observed by Schwab et al. [Nature (London) 404, 974 (2000)] in terms of the scattering of thermal phonons off surface roughness using a scalar model for the elastic waves. Our analysis shows that the thermal conductance depends on two roughness parameters: the roughness amplitude \u03b4 and the correlation length a. At sufficiently low temperatures the ratio of the conductance to the universal value decreases quadratically with temperature at a rate proportional to \u03b42a. Values of \u03b4 equal to 22% and a equal to about 75% of the width of the conduction pathway give a good fit to the data.", "doi": "10.1103/PhysRevB.63.184306", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "2001-05-01", "series_number": "18", "volume": "63", "issue": "18", "pages": "Art. No. 184306" }, { "id": "authors:h4r9s-g1m11", "collection": "authors", "collection_id": "h4r9s-g1m11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpre01", "type": "article", "title": "Finite size scaling of domain chaos", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Louie", "given_name": "M.", "clpid": "Louie-M" }, { "family_name": "Meiron", "given_name": "D.", "orcid": "0000-0003-0397-3775", "clpid": "Meiron-D-I" } ], "abstract": "Numerical studies of the domain chaos state in a model of rotating Rayleigh-B\u00e9nard convection suggest that finite size effects may account for the discrepancy between experimentally measured values of the correlation length and the predicted divergence near onset.", "doi": "10.1103/PhysRevE.63.045201", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "2001-04-01", "series_number": "4", "volume": "63", "issue": "4", "pages": "Art. No. 045201" }, { "id": "authors:ycxmb-4nf24", "collection": "authors", "collection_id": "ycxmb-4nf24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150330-100637705", "type": "article", "title": "Nonlinear dynamics: Chaos in space and time", "author": [ { "family_name": "Gollub", "given_name": "J. P.", "clpid": "Gollub-J-P" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Many aspects of nature are essentially unpredictable over the long term, even when quantum effects are completely absent. A frequent cause is chaotic dynamics, which can arise when the rates of change of important dynamical variables depend in a nonlinear fashion on the variables themselves. Famous examples are the unpredictable behaviour of the weather and of populations of organisms. Chaotic temporal dynamics has now been documented in practically every scientific discipline \u2014 the fascination for both scientists and the public is that chaos helps us understand what can be predicted about the systems we study, and what cannot. But our understanding of chaos in most systems found in nature, which are complex in both space and time, is relatively limited. On page 733 of this issue, Egolf et al. analyse the chaotic space and time variations of a heated fluid in a way that could eventually be applied to natural systems such as the weather.", "doi": "10.1038/35008168", "issn": "0028-0836", "publisher": "Nature Publishing Group", "publication": "Nature", "publication_date": "2000-04-13", "series_number": "6779", "volume": "404", "issue": "6779", "pages": "710" }, { "id": "authors:n6ag9-tyn90", "collection": "authors", "collection_id": "n6ag9-tyn90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:TANprb00", "type": "article", "title": "Ballistic spin transport in a two-dimensional electron gas", "author": [ { "family_name": "Tang", "given_name": "H. X.", "clpid": "Tang-Hongxing" }, { "family_name": "Monzon", "given_name": "F. G.", "clpid": "Monzon-F-G" }, { "family_name": "Lifshitz", "given_name": "Ron", "orcid": "0000-0002-8829-5506", "clpid": "Lifshitz-R" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "We explore electrically injected, spin-polarized transport in a ballistic two-dimensional electron gas. We augment the B\u00fcttiker-Landauer picture with a simple, but realistic model for spin-selective contacts to describe multimode reservoir-to-reservoir transport of ballistic spin-1/2 particles. Clear and unambiguous signatures of spin transport are established in this regime, for the simplest measurement configuration that demonstrates them directly. These effects originate from spin precession of ballistic carriers; they exhibit a strong dependence upon device geometry, and vanish in the diffusive limit. Our results have important implications for prospective \"spin transistor\" devices.", "doi": "10.1103/PhysRevB.61.4437", "issn": "0163-1829", "publisher": "American Physical Society", "publication": "Physical Review B", "publication_date": "2000-02-15", "series_number": "7", "volume": "61", "issue": "7", "pages": "4437-4440" }, { "id": "authors:hms56-r7h78", "collection": "authors", "collection_id": "hms56-r7h78", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SETpre99", "type": "article", "title": "Numerical bifurcation diagram for the two-dimensional boundary-fed chlorine-dioxide\u2013iodine\u2013malonic-acid system", "author": [ { "family_name": "Setayeshgar", "given_name": "S.", "clpid": "Setayeshgar-S" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We present a numerical solution of the chlorine-dioxide\u2013iodine\u2013malonic-acid reaction-diffusion system in two dimensions in a boundary-fed system using a realistic model. The bifurcation diagram for the transition from nonsymmetry-breaking structures along boundary feed gradients to transverse symmetry-breaking patterns in a single layer is numerically determined. We find this transition to be discontinuous. We make a connection with earlier results and discuss prospects for future work.", "doi": "10.1103/PhysRevE.59.4258", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "1999-04-01", "series_number": "4", "volume": "59", "issue": "4", "pages": "4258-4264" }, { "id": "authors:fh04m-rj316", "collection": "authors", "collection_id": "fh04m-rj316", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SETpre98", "type": "article", "title": "Turing instability in a boundary-fed system", "author": [ { "family_name": "Setayeshgar", "given_name": "S.", "clpid": "Setayeshgar-S" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The formation of localized structures in the chlorine dioxide-idodine-malonic acid (CDIMA) reaction-diffusion system is investigated numerically using a realistic model of this system. We analyze the one-dimensional patterns formed along the gradients imposed by boundary feeds, and study their linear stability to symmetry-breaking perturbations (Turing instability) in the plane transverse to these gradients. We establish that an often-invoked simple local linear analysis that neglects longitudinal diffusion is inappropriate for predicting the linear stability of these patterns. Using a fully nonuniform analysis, we investigate the structure of the patterns formed along the gradients and their stability to transverse Turing pattern formation as a function of the values of two control parameters: the malonic acid feed concentration and the size of the reactor in the dimension along the gradients. The results from this investigation are compared with existing experiments.", "doi": "10.1103/PhysRevE.58.4485", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "1998-10-01", "series_number": "4", "volume": "58", "issue": "4", "pages": "4485-4500" }, { "id": "authors:5qjmk-7d355", "collection": "authors", "collection_id": "5qjmk-7d355", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190628-091042894", "type": "article", "title": "Heat transport in mesoscopic systems", "author": [ { "family_name": "Angelescu", "given_name": "D. E.", "clpid": "Angelescu-D-E" }, { "family_name": "Cross", "given_name": "M", "clpid": "Cross-M-C" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "Phonon heat transport in mesoscopic systems is investigated using methods analogous to the Landauer description of electrical conductance. A 'universal heat conductance' expression that depends on the properties of the conducting pathway only through the mode cutoff frequencies is derived. Corrections due to reflections at the junction between the thermal body and the conducting bridge are found to be small except at very low temperatures where only the lowest few bridge modes are excited. Various non-equilibrium phonon distributions are studied: a narrow band distribution leads to clear steps in the cooling curve, analogous to the quantized resistance values in narrow wires, but a thermal distribution is too broad to show such features.", "doi": "10.1006/spmi.1997.0561", "issn": "0749-6036", "publisher": "Elsevier", "publication": "Superlattices and Microstructures", "publication_date": "1998-03", "series_number": "3-4", "volume": "23", "issue": "3-4", "pages": "673-689" }, { "id": "authors:ay7ns-3jd02", "collection": "authors", "collection_id": "ay7ns-3jd02", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GRIpre98", "type": "article", "title": "Controlling physical systems with symmetries", "author": [ { "family_name": "Grigoriev", "given_name": "R. O.", "clpid": "Grigoriev-R-O" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Symmetry properties of the evolution equation and the state to be controlled are shown to determine the basic features of the linear control of unstable orbits. In particular the selection of control parameters and their minimal number are determined by the irreducible representations of the symmetry group of the linearization about the orbit to be controlled. We use the general results to demonstrate the effect of symmetry on the control of two sample physical systems: a coupled map lattice and a particle in a symmetric potential.", "doi": "10.1103/PhysRevE.57.1550", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "1998-02-01", "series_number": "2", "volume": "57", "issue": "2", "pages": "1550-1554" }, { "id": "authors:e0mvw-3xb52", "collection": "authors", "collection_id": "e0mvw-3xb52", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:ENGjosab97", "type": "article", "title": "Amplitude-equation formalism for four-wave-mixing geometry with transmission gratings", "author": [ { "family_name": "Engin", "given_name": "D.", "clpid": "Engin-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Yariv", "given_name": "A.", "clpid": "Yariv-A" } ], "abstract": "An amplitude equation is derived for a four-wave-mixing geometry with nearly counterpropagating, mutually incoherent, nondiffracting pump beams, spatially overlapping in a photorefractive material with a nonlocal response. This equation extends the earlier linear two-dimensional theory to the weakly nonlinear regime. The analysis also starts from a more complete equation for the photorefractive effect, which leads to the prediction of novel effects especially apparent in the nonlinear regime. Precise predictions for the spatiotemporal behavior of the grating amplitude in the nonlinear regime are presented. The range of validity of the amplitude equation is studied. The characteristics of the instability in the nonlinear regime are analyzed through a front-selection analysis.", "issn": "0740-3224", "publisher": "Optical Society of America", "publication": "Journal of the Optical Society of America B", "publication_date": "1997-12", "series_number": "12", "volume": "14", "issue": "12", "pages": "3349-3362" }, { "id": "authors:yvrak-3vp44", "collection": "authors", "collection_id": "yvrak-3vp44", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GRIprl97", "type": "article", "title": "Pinning control of spatiotemporal chaos", "author": [ { "family_name": "Grigoriev", "given_name": "R. O.", "clpid": "Grigoriev-R-O" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Schuster", "given_name": "H. G.", "clpid": "Schuster-H-G" } ], "abstract": "Linear control theory is used to develop an improved localized control scheme for spatially extended chaotic systems, which is applied to a coupled map lattice as an example. The optimal arrangement of the control sites is shown to depend on the symmetry properties of the system, while their minimal density depends on the strength of noise in the system. The method is shown to work in any region of parameter space and requires a significantly smaller number of controllers compared to the method proposed earlier by Hu and Qu [Phys. Rev. Lett. 72, 68 (1994)]. A nonlinear generalization of the method for a 1D lattice is also presented.", "doi": "10.1103/PhysRevLett.79.2795", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1997-10-13", "series_number": "15", "volume": "79", "issue": "15", "pages": "2795 -2798" }, { "id": "authors:s4apm-jfs41", "collection": "authors", "collection_id": "s4apm-jfs41", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CHEpre97", "type": "article", "title": "Complete phase diagram for coherent vortex formation in a two-dimensional inviscid fluid in an annulus", "author": [ { "family_name": "Chen", "given_name": "Peilong", "clpid": "Chen-P" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "By calculating the two-dimensional solutions and the second-order bifurcation analysis of the mean field equations of a two-dimensional ideal fluid, we are able to complete the phase diagram of coherent vortex formation in an annulus, which we have partially calculated in a previous paper [Phys. Rev. E 50, 2022 (1994)]. The complete phase diagram demonstrates the advantage of the statistical mechanics approach to determine the final equilibrium directly from the initial condition.", "doi": "10.1103/PhysRevE.56.2284", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "1997-08-01", "series_number": "2", "volume": "56", "issue": "2", "pages": "2284-2286" }, { "id": "authors:2a5wv-65937", "collection": "authors", "collection_id": "2a5wv-65937", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GRIchaos97", "type": "article", "title": "Dynamics of coupled maps with a conservation law", "author": [ { "family_name": "Grigoriev", "given_name": "R. O.", "clpid": "Grigoriev-R-O" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "A particularly simple model belonging to a wide class of coupled maps which obey a local conservation law is studied. The phase structure of the system and the types of the phase transitions are determined. It is argued that the structure of the phase diagram is robust with respect to mild violations of the conservation law. Critical exponents possibly determining a new universality class are calculated for a set of independent order parameters. Numerical evidence is produced suggesting that the singularity in the density of Lyapunov exponents at lambda = 0 is a reflection of the singularity in the density of Fourier modes (a \"Van Hove\" singularity) and disappears if the conservation law is broken. Applicability of the Lyapunov dimension to the description of spatiotemporal chaos in a system with a conservation law is discussed.", "doi": "10.1063/1.166222", "issn": "1054-1500", "publisher": "Chaos", "publication": "Chaos", "publication_date": "1997-06-01", "series_number": "2", "volume": "7", "issue": "2", "pages": "311-330" }, { "id": "authors:tqxwz-v2q44", "collection": "authors", "collection_id": "tqxwz-v2q44", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120208-070847304", "type": "book_section", "title": "Transition to an oscillator for double phase-conjugate mirror", "book_title": "Summaries of Papers Presented at the 1996 Quantum Electronics and Laser Science Conference", "author": [ { "family_name": "Engin", "given_name": "D.", "clpid": "Engin-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Yariv", "given_name": "A.", "clpid": "Yariv-A" }, { "family_name": "Segev", "given_name": "M.", "clpid": "Segev-M" } ], "abstract": "Summary form only given. Some of the novel quantified characteristics for double phase conjugate mirrors are analysed including the effects of the nonlinearity on the critical dynamics (approach to saturation) and on the spatial distribution of the grating (large scale distortion of the beams and conjugation fidelity) and sensitivity to noise (seeding). The approach used also clarifies the question of linear instability and predicts a new transition to an oscillatory regime.", "isbn": "1-55752-444-0", "publisher": "IEEE", "place_of_publication": "Washington, D.C.", "publication_date": "1996-06", "pages": "118-119" }, { "id": "authors:cynyw-7zy48", "collection": "authors", "collection_id": "cynyw-7zy48", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CHEpre96", "type": "article", "title": "Statistical two-vortex equilibrium and vortex merger", "author": [ { "family_name": "Chen", "given_name": "Peilong", "clpid": "Chen-P" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Two-vortex solutions from the mean field equations respecting all conservation laws of the Euler equations in two dimensions are calculated in a unit disk. These states are compared to symmetric and off-center single-vortex solutions. A thermodynamic stability analysis is used to determine their stability and a critical separation for stable two-vortex states is found from the crossover to instability. This could explain the merging critical ratios observed in numerical simulations and experiments.", "doi": "10.1103/PhysRevE.53.R3032", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "1996-04-01", "series_number": "4", "volume": "53", "issue": "4", "pages": "R3032-R3035" }, { "id": "authors:d6ptv-8y953", "collection": "authors", "collection_id": "d6ptv-8y953", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SCHWprb95", "type": "article", "title": "Perturbation of tunneling processes by mechanical degrees of freedom in mesoscopic junctions", "author": [ { "family_name": "Schwabe", "given_name": "N. F.", "clpid": "Schwabe-N-F" }, { "family_name": "Cleland", "given_name": "A. N.", "clpid": "Cleland-N-F" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Roukes", "given_name": "M. L.", "orcid": "0000-0002-2916-6026", "clpid": "Roukes-M-L" } ], "abstract": "We investigate the perturbation in the tunneling current caused by nonadiabatic mechanical motion in a mesoscopic tunnel junction. A theory introduced by Caroli et al. is used to evaluate second-order self-energy corrections for this nonequilibrium situation lacking translational invariance. Inelastic signatures of the mechanical degrees of freedom are found in the current-voltage I(V) characteristics. These give rise to sharp features in the derivative spectrum, d2I/dV2.", "issn": "0163-1829", "publisher": "American Physical Society", "publication": "Physical Review B", "publication_date": "1995-11-01", "series_number": "17", "volume": "52", "issue": "17", "pages": "12911-12920" }, { "id": "authors:6fxjf-8tr69", "collection": "authors", "collection_id": "6fxjf-8tr69", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprl95b", "type": "article", "title": "Domain Coarsening in Systems Far from Equilibrium", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Meiron", "given_name": "D. I.", "orcid": "0000-0003-0397-3775", "clpid": "Meiron-D-I" } ], "abstract": "The growth of domains of stripes evolving from random initial conditions is studied in numerical simulations of models of systems far from equilibrium such as Rayleigh-B\u00e9nard convection. The size of the domains deduced from the inverse width of the Fourier spectrum is found to scale as t^1/5 for both potential and nonpotential models. The morphology of the domains and the defect structures are, however, quite different in the two cases, and evidence is presented for a second length scale in the nonpotential case growing as t^1/2.", "doi": "10.1103/PhysRevLett.75.2152", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1995-09-11", "series_number": "11", "volume": "75", "issue": "11", "pages": "2152-2155" }, { "id": "authors:vcqb2-vy355", "collection": "authors", "collection_id": "vcqb2-vy355", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprl95a", "type": "article", "title": "Defect Dynamics for Spiral Chaos in Rayleigh-B\u00e9nard Convection", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Tu", "given_name": "Yuhai", "clpid": "Tu-Y" } ], "abstract": "A theory of the novel spiral chaos state recently observed in Rayleigh-B\u00e9nard convection is proposed in terms of the importance of invasive defects, i.e., defects that through their intrinsic dynamics expand to take over the system. The motion of well developed spiral defects is dominated by wave vector frustration, rather than a rotational motion driven by a vertical vorticity field. This leads to a continuum of spiral frequencies, and a spiral may rotate in either sense depending on the wave vector of its local environment. Results of extensive numerical work on equations modeling the convection system provide confirmation of these ideas.", "doi": "10.1103/PhysRevLett.75.834", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1995-07-31", "series_number": "5", "volume": "75", "issue": "5", "pages": "834-837" }, { "id": "authors:7crr6-zcv03", "collection": "authors", "collection_id": "7crr6-zcv03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROchaos94", "type": "article", "title": "Chaotic domains: A numerical investigation", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Meiron", "given_name": "D.", "orcid": "0000-0003-0397-3775", "clpid": "Meiron-D-I" }, { "family_name": "Tu", "given_name": "Yuhai", "clpid": "Tu-Y" } ], "abstract": "We study the chaotic domain state in rotating convection using a model equation that allows for a continuous range of roll orientations as in the experimental system. Methods are developed for extracting the domain configuration from the resulting patterns that should be applicable to a wide range of domain states. Comparison with the truncated three mode amplitude equation description is made.", "doi": "10.1063/1.166038", "issn": "1054-1500", "publisher": "Chaos", "publication": "Chaos", "publication_date": "1994-12-01", "series_number": "4", "volume": "4", "issue": "4", "pages": "607-619" }, { "id": "authors:1vs3a-t1g43", "collection": "authors", "collection_id": "1vs3a-t1g43", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CHEpre94", "type": "article", "title": "Phase diagram for coherent vortex formation in the two-dimensional inviscid fluid in circular geometries", "author": [ { "family_name": "Chen", "given_name": "Peilong", "clpid": "Chen-P" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We calculate the equilibrium states of a two-dimensional inviscid fluid in disk and annular geometries using the mean field equations that respect all conservation laws of the Euler equations. Axisymmetric vorticity distributions and their bifurcations to asymmetric solutions are calculated for a wide range of system parameters. Approximate zero-temperature compact vortices are also constructed and compared to symmetric states. From these results, the parameter ranges leading to the formation of a coherent vortex by the long time evolution from given initial conditions can be predicted without knowledge of the intervening dynamics. Applications to models of Jupiter's great red spot and experiments on one-component plasmas in a magnetic field are discussed.", "doi": "10.1103/PhysRevE.50.2022", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "1994-09-01", "series_number": "3", "volume": "50", "issue": "3", "pages": "2022-2029" }, { "id": "authors:s9afb-f6m66", "collection": "authors", "collection_id": "s9afb-f6m66", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CHEnonlin94", "type": "article", "title": "Pattern formation in finite size non-equilibrium systems and models of morphogenesis", "author": [ { "family_name": "Chen", "given_name": "Yih-Yuh", "clpid": "Chen-Yih-Yuh" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Two canonical pattern forming systems, the Rayleigh-Benard convection and the Turing mechanism for biological pattern formation, are compared. The similarity and fundamental differences in the mathematical structure of the two systems are addressed, with special emphasis on how the linear onset of patterns is affected by the finite size and the boundary conditions. Our analysis is facilitated by continuously varying the boundary condition, from one that admits simple algebraic solution of the problem but is unrealistic to another which is physically realizable. Our investigation shows that the size dependence of the convection problem can be considered generic, in the sense that for the majority of boundary conditions the same trend is to be observed, while for the corresponding Turing mechanism one will rely crucially on the assumed boundary conditions to ensure that a particular sequence of patterns be picked up as the system grows in size. This suggests that, although different systems might exhibit similar pattern forming features, it is still possible to distinguish them by characteristics which are specific to the individual models.", "doi": "10.1088/0951-7715/7/4/001", "issn": "0951-7715", "publisher": "Nonlinearity", "publication": "Nonlinearity", "publication_date": "1994-07", "series_number": "4", "volume": "1994", "issue": "4", "pages": "1125-1132" }, { "id": "authors:49cng-7mf96", "collection": "authors", "collection_id": "49cng-7mf96", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150123-091832757", "type": "article", "title": "Spatiotemporal Chaos", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Hohenberg", "given_name": "P. C.", "clpid": "Hohenberg-P-C" } ], "abstract": "The term \"chaos\" denotes persistent irregular behavior of a deterministic system (that is, one in which externally applied noise can be neglected). Much of the work on chaos of the last 15 years has dealt with systems that could be represented by a small number of degrees of freedom, such as the logistic map u_(t+1) = ru_t(1 - u_t) or the famous Lorenz model involving nonlinear differential equations for three coupled dynamical variables. Remarkably, at least near enough to the threshold of chaos, continuous experimental systems such as stirred chemical reactors or flows in certain fluid cells can be represented by such models.", "doi": "10.1126/science.263.5153.1569", "issn": "0036-8075", "publisher": "American Association for the Advancement of Science", "publication": "Science", "publication_date": "1994-03-18", "series_number": "5153", "volume": "263", "issue": "5153", "pages": "1569-1570" }, { "id": "authors:kpr6z-ndm09", "collection": "authors", "collection_id": "kpr6z-ndm09", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROrmp93", "type": "article", "title": "Pattern formation outside of equilibrium", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Hohenberg", "given_name": "P. C.", "clpid": "Hohenberg-P-C" } ], "abstract": "A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency \u03c90 of the instability. Type Is systems (\u03c90=0, q0\u22600) are stationary in time and periodic in space; type IIIo systems (\u03c90\u22600, q0=0) are periodic in time and uniform in space; and type Io systems (\u03c90\u22600, q0\u22600) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via \"amplitude equations,\" whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the \"phase equation\" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing \"real pattern effects\" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying \"temporal chaos,\" where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-B\u00e9nard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.", "doi": "10.1103/RevModPhys.65.851", "issn": "0034-6861", "publisher": "Reviews of Modern Physics", "publication": "Reviews of Modern Physics", "publication_date": "1993-07-01", "series_number": "3", "volume": "65", "issue": "3", "pages": "851-1112" }, { "id": "authors:1435p-rtp27", "collection": "authors", "collection_id": "1435p-rtp27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KUOpre93", "type": "article", "title": "Traveling-wave wall states in rotating Rayleigh-B\u00e9nard convection", "author": [ { "family_name": "Kuo", "given_name": "E. Y.", "clpid": "Kuo-E-Y" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We show that the wall-localized convection states observed in Rayleigh-B\u00e9nard convection in rotating cylindrical cells can be explained in terms of a geometry-independent traveling-wave wall state. We calculate the onset Rayleigh number, frequency, and wave number of such a state, as well as its amplitude equation. We also study the large-rotation-rate asymptotic behavior and the small-rotation-rate limit.", "doi": "10.1103/PhysRevE.47.R2245", "issn": "1063-651X", "publisher": "Physical Review E", "publication": "Physical Review E", "publication_date": "1993-04-01", "series_number": "4", "volume": "47", "issue": "4", "pages": "R2245-R2248" }, { "id": "authors:5kz6v-3eg77", "collection": "authors", "collection_id": "5kz6v-3eg77", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:TUYprl92", "type": "article", "title": "Chaotic domain structure in rotating convection", "author": [ { "family_name": "Tu", "given_name": "Yuhai", "clpid": "Tu-Y" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Rotating convection near onset is studied using a three mode model but with the inclusion of spatial dependence. An apparently chaotic dynamics is found, consistent with experiment, without the necessity of adding ad hoc noise terms as in previous models. There is no phase transition approaching the point of linear instability of straight rolls from the disordered side, and the characteristic domain size and switching time remain finite here. This behavior is discussed in terms of the velocity of walls between the different domains.", "doi": "10.1103/PhysRevLett.69.2515", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1992-10-26", "series_number": "17", "volume": "69", "issue": "17", "pages": "2515-2518" }, { "id": "authors:xyx5f-fqv80", "collection": "authors", "collection_id": "xyx5f-fqv80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BOUpra92", "type": "article", "title": "Reflection of traveling waves near the onset of binary-fluid convection", "author": [ { "family_name": "Bourzutschky", "given_name": "M. S.", "clpid": "Bourzutschky-M-S" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The reflection coefficient of linear traveling waves in binary-fluid convection is calculated for the experimental situation of rigid, impermeable boundaries. Results for a range of parameters of interest to experiments in ethanol-water mixtures are displayed and compared with experiment.", "doi": "10.1103/PhysRevA.45.R8317", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1992-06-15", "series_number": "12", "volume": "45", "issue": "12", "pages": "R8317-R8320" }, { "id": "authors:z6mhz-0k482", "collection": "authors", "collection_id": "z6mhz-0k482", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BOUchaos92", "type": "article", "title": "Coupled map models for chaos in extended systems", "author": [ { "family_name": "Bourzutschky", "given_name": "M. S.", "clpid": "Bourzutschky-M-S" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Coupled maps with conserved quantities are introduced as models for chaos in extended systems. The long-wavelength limit of a simple one-dimensional example is investigated in detail. A Langevin equation for this system is derived, and the applicability of the fluctuation-dissipation theorem is discussed. Several numerical experiments to investigate possible thermodynamic properties are described.", "doi": "10.1063/1.165905", "issn": "1054-1500", "publisher": "Chaos", "publication": "Chaos", "publication_date": "1992-04-01", "series_number": "2", "volume": "2", "issue": "2", "pages": "173-181" }, { "id": "authors:zbwkm-0zd57", "collection": "authors", "collection_id": "zbwkm-0zd57", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:MILpra92", "type": "article", "title": "Statistical mechanics, Euler's equation, and Jupiter's Red Spot", "author": [ { "family_name": "Miller", "given_name": "Jonathan", "clpid": "Miller-J" }, { "family_name": "Weichman", "given_name": "Peter B.", "clpid": "Weichman-P-B" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We construct a statistical-mechanical treatment of equilibrium flows in the two-dimensional Euler fluid which respects all conservation laws. The vorticity field is fundamental, and its long-range Coulomb interactions lead to an exact set of nonlinear mean-field equations for the equilibrium state. The equations depend on an infinite set of parameters, in one-to-one correspondence with the infinite set of conserved variables. We illustrate the equations by solving them numerically in simple cases. In more complicated cases we use Monte Carlo techniques, with the eventual aim of detailed comparison with the Red Spot dynamical simulations of Marcus: Preliminary efforts show good agreement. We review previous literature on two-dimensional (2D) Euler flow in light of our theory. In particular, we derive the Kraichnan energy-enstrophy theory and the Lundgren and Pointin point-vortex mean-field theory as special cases of our own. Our techniques may be generalized to a number of other Coulomb-like Hamiltonian systems with an infinite number of conservation laws, including some in higher dimensions. For example, we rederive Lynden-Bell's theory of stellar-cluster formation, as well as the Debye-Huckel theory of electrolytes. Our results may also be applicable to cylindrically bound guiding-center plasmas, which under idealized conditions provide another realization of 2D Euler flow.", "doi": "10.1103/PhysRevA.45.2328", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1992-02-15", "series_number": "4", "volume": "45", "issue": "4", "pages": "2328-2359" }, { "id": "authors:adg44-3eg25", "collection": "authors", "collection_id": "adg44-3eg25", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FRIprl89", "type": "article", "title": "Zero-field magnetic order in the boundary layers of 3He on Grafoil", "author": [ { "family_name": "Friedman", "given_name": "L. J.", "clpid": "Friedman-L-J" }, { "family_name": "Thomson", "given_name": "A. L.", "clpid": "Thomson-A-L" }, { "family_name": "Gould", "given_name": "C. M.", "clpid": "Gould-C-M" }, { "family_name": "Bozler", "given_name": "H. M.", "clpid": "Bozler-H-M" }, { "family_name": "Weichman", "given_name": "P. B.", "clpid": "Weichman-P-B" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The low-field NMR spectra of 3He boundary layers on exfoliated graphite show collective modes for T<1 mK. We measure the amplitude and frequency for these modes with the static H0 field applied parallel to the graphite planes and varying continuously between 0 and 15 G. One of the modes extrapolates to a nonzero frequency and amplitude as the field is dropped to zero. We interpret these nonzero intercepts as an indication of zero-field magnetic order in the 3He boundary layers.", "doi": "10.1103/PhysRevLett.62.1635", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1989-04-03", "series_number": "14", "volume": "62", "issue": "14", "pages": "1635-1638" }, { "id": "authors:jmf77-4p756", "collection": "authors", "collection_id": "jmf77-4p756", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:AVRprb89", "type": "article", "title": "Integer charge transport in Josephson junctions", "author": [ { "family_name": "Avron", "given_name": "J. E.", "clpid": "Avron-J-E" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We describe a macroscopic demonstration of quantized adiabatic transport coefficients using superconductors coupled by Josephson junctions. The transport coefficients relate charges and currents to changes in potential differences and loop fluxes. They have geometrical significance and quantized average related to first Chern characters.", "doi": "10.1103/PhysRevB.39.756", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "1989-01-01", "series_number": "1", "volume": "39", "issue": "1", "pages": "756-758" }, { "id": "authors:0kwsq-qmc91", "collection": "authors", "collection_id": "0kwsq-qmc91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra88c", "type": "article", "title": "Structure of nonlinear traveling-wave states in finite geometries", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Numerical simulation of coupled amplitude equations is used to investigate the effect of the wave propagation on the one-dimensional spatial structure of nonlinear wave states in finite geometries. The work is motivated by experiments on oscillatory convection in binary fluid convection. Predictions of confined states, temporally modulated confined states, and other dynamic states are discussed and compared with experiment.", "doi": "10.1103/PhysRevA.38.3593", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1988-10-01", "series_number": "7", "volume": "38", "issue": "7", "pages": "3593-3600" }, { "id": "authors:c9es1-v2p91", "collection": "authors", "collection_id": "c9es1-v2p91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra88b", "type": "article", "title": "Existence of a codimension-2 point at the threshold of binary-fluid convection between rigid, impermeable boundaries", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Kim", "given_name": "K.", "clpid": "Kim-K" } ], "abstract": "Numerical calculation of the linear instability to oscillatory and stationary convection in binary-fluid mixtures between rigid, impermeable boundaries shows that the critical wave number and frequency of the oscillatory solution jump discontinuously as the threshold Rayleigh number is followed as a function of fluid parameters, contrary to a recent claim of B. J. A. Zielinska and H. R. Brand [Phys. Rev. A 35, 4349 (1987)]. A degenerate codimension-2 point is found only if the wave numbers of the onset solutions are constrained.", "doi": "10.1103/PhysRevA.38.529", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1988-07-01", "series_number": "1", "volume": "38", "issue": "1", "pages": "529-530" }, { "id": "authors:q2f1m-27s61", "collection": "authors", "collection_id": "q2f1m-27s61", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GREprl88", "type": "article", "title": "Mean flows and the onset of chaos in large-cell convection", "author": [ { "family_name": "Greenside", "given_name": "H. S.", "clpid": "Greenside-H-S" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Coughran", "given_name": "W. M., Jr.", "clpid": "Coughran-W-M-Jr" } ], "abstract": "Numerical simulations of two-dimensional model equations show that a coupling between amplitude and vertical-vorticity fields allows chaotic flows near the onset of Rayleigh-B\u00e9nard convection in large-aspect-ratio domains. In cylindrical cells, mean flows arising from this coupling lead to a chaotic nucleation of dislocations that is remarkably similar to recent observations in convection experiments.", "doi": "10.1103/PhysRevLett.60.2269", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1988-05-30", "series_number": "22", "volume": "60", "issue": "22", "pages": "2269-2272" }, { "id": "authors:4prz8-tts17", "collection": "authors", "collection_id": "4prz8-tts17", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra88a", "type": "article", "title": "Linear instability and the codimension-2 region in binary fluid convection between rigid impermeable boundaries", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Kim", "given_name": "Kihong", "clpid": "Kim-K" } ], "abstract": "The parameters of the linear instability to oscillatory convection (critical Rayleigh number, onset frequency, and others) are calculated for the experimentally common situation of rigid, impermeable boundaries, both near and away from the degenerate (codimension-2) bifurcation with stationary convection. This gives all linear coefficients of the standard and degenerate-amplitude equations. The small-Lewis-number limit is explicitly calculated. Wave-number and frequency jumps are confirmed in the vicinity of the codimension-2 point.", "doi": "10.1103/PhysRevA.37.3909", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1988-05-15", "series_number": "10", "volume": "37", "issue": "10", "pages": "3909-3920" }, { "id": "authors:rfzp0-j4c30", "collection": "authors", "collection_id": "rfzp0-j4c30", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20200930-113055493", "type": "book_section", "title": "Theoretical methods in pattern formation in Physics, Chemistry and Biology", "book_title": "Far from Equilibrium Phase Transitions", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "contributor": [ { "family_name": "Garrido", "given_name": "Luis", "clpid": "Garrido-L" } ], "abstract": "In these lectures some simple ideas common to theories of pattern formation in diverse areas of Physics, Chemistry and Biology are reviewed. The unifying theme of these theoretical models is the existence of an instability (bifurcation) in a system driven far from equilibrium to a state with spatial variation at a non-zero wave vector. The basic pattern forming tendency is understood by a linear stability analysis of the spatially uniform system. The subsequent saturation of the exponential growth by non-linearities together with boundary effects (the influence of the finite size of the domain) leads to the complexity and subtlety in the understanding of non-equilibrium structures. Some elements of the theory of linear instability, non-linearity and boundary effects will be briefly reviewed in these lectures. The emphasis is on common features of the theoretical modeling of the phenomena. No attempt will be made to assess the detailed applicability of these models to the different fields, or the main aims of the study of structure formation in the different fields. There are clearly great differences between the carefully controlled laboratory models of the physicist studying hydrodynamic instabilities the chemist studying transient evolution of complicated chemical reactions, and the in vivo observations of the biologist studying developing organisms, differences in the ability to control the system, the understanding of the underlying microscopic processes, and particularly the goals of the endeavor. At the current elementary understanding of the general phenomena, these disparities are not very evident in the theoretical modeling. \n\nIn the second part of the lectures I will report on two examples of the application of some of the above methods. \n\nThese lectures are derived from a longer forthcoming review with P.C. Hohenberg. A general overview has been presented elsewhere (Hohenberg and Cross, 1986): The present lectures provide a more detailed presentation of a narrower range of topics.", "doi": "10.1007/3-540-50643-8_33", "isbn": "9783540506430", "publisher": "Springer Berlin Heidelberg", "place_of_publication": "Berlin, Heidelberg", "publication_date": "1988", "pages": "45-74" }, { "id": "authors:96cyq-b7m98", "collection": "authors", "collection_id": "96cyq-b7m98", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:OSHprl87", "type": "article", "title": "Magnetostatic modes in highly polarized solid helium-three", "author": [ { "family_name": "Osheroff", "given_name": "D. D.", "clpid": "Osheroff-D-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We report the observation of an unusual series of spin-wave modes seen in continuous-wave NMR measurements on long cylindrical crystals of highly polarized solid 3He. When a magnetic field and field gradient of appropriate magnitude are placed along the cylinder axis, the NMR spectrum breaks up into a series of discrete peaks whose spacing is geometric in mode number, with the spacing going to zero at an accumulation point. We show that this behavior is consistent with magnetostatic waves in the highly polarized medium.", "doi": "10.1103/PhysRevLett.59.94", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1987-07-06", "series_number": "1", "volume": "59", "issue": "1", "pages": "94-97" }, { "id": "authors:qy2j2-4ms84", "collection": "authors", "collection_id": "qy2j2-4ms84", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210123-172818647", "type": "book_section", "title": "An introduction to pattern formation in nonequilibrium systems", "book_title": "Fluctuations and Stochastic Phenomena in Condensed Matter", "author": [ { "family_name": "Hohenberg", "given_name": "P. C.", "clpid": "Hohenberg-P-C" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "contributor": [ { "family_name": "Garrido", "given_name": "L.", "clpid": "Garrido-L" } ], "abstract": "Patterns which arise in various nonequilibrium systems are discussed from an elementary but unified point of view. The phenomena of interest range from hydrodynamic flows (Rayleigh- Benard convection, Taylor-Couette flow, parametric waves) to chemical instabilities and morphogenesis in biological systems. A unifying feature is provided by linear stability theory which leads to a classification of patterns depending on the spatial and temporal scales of the instability. Near the instability point a universal amplitude equation is derived and used to elucidate many elementary properties of the system. The simplest solutions of this equation, which we call ideal patterns, are those with maximal symmetry. Some of their properties can be described also away from threshold by means of phase equations or more abstractly in terms of topological concepts. Real patterns, which we consider next, differ from ideal ones through the influence of boundaries and the loss of symmetry caused by spatial disorder. The simplest example of spatial disorder is a single defect in an otherwise regular pattern. Our treatment of real patterns focuses almost exclusively on Rayleigh-Benard convection since this is by far the most studied system. The importance of boundaries and defects is illustrated in the study of wavevector selection and in the description of pattern dynamics near threshold. Numerical and analytical studies of model equations have proved useful in the exploration of these difficult problems, but our present level of understanding is still far from complete.", "doi": "10.1007/3-540-17206-8_3", "isbn": "9783540172062", "publisher": "Springer Berlin Heidelberg", "place_of_publication": "Berlin", "publication_date": "1987", "pages": "55-92" }, { "id": "authors:69trb-1yg28", "collection": "authors", "collection_id": "69trb-1yg28", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FRIprl86", "type": "article", "title": "Detection of ferromagnetic domains in a two-dimensional nuclear-spin system", "author": [ { "family_name": "Friedman", "given_name": "L. J.", "clpid": "Friedman-L-J" }, { "family_name": "Ytterboe", "given_name": "S. N.", "clpid": "Ytterboe-S-N" }, { "family_name": "Bozler", "given_name": "H. M.", "clpid": "Bozler-H-M" }, { "family_name": "Thomson", "given_name": "A. L.", "clpid": "Thomson-A-L" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We observe magnetic ordering of the 3He spins at the surface of exfoliated graphite. In very low applied magnetic fields (\u22643 G) we find new modes in the NMR resonance spectrum with the amplitude of these lines increasing sharply below 0.8 mK. We interpret our results in terms of a ferromagnetically aligned spin system, and conclude that there must be an anisotropy in the graphite planes.", "doi": "10.1103/PhysRevLett.57.2943", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1986-12-08", "series_number": "23", "volume": "57", "issue": "23", "pages": "2943-2946" }, { "id": "authors:0jj34-9cx60", "collection": "authors", "collection_id": "0jj34-9cx60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprl86", "type": "article", "title": "Traveling and Standing Waves in Binary-Fluid Convection in Finite Geometries", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The effect of finite geometry on the competition between traveling waves and standing waves in systems with a Hopf bifurcation to a state with spatial structure is considered in the linear and weakly nonlinear regimes. The spatial structure observed by Kolodner et al. in binary-fluid convection is explained in terms of the reflection of the linear traveling waves. The reflection coefficient is calculated, and is found to go to zero as the frequency of the waves becomes small. The pattern expected in a saturated nonlinear state is discussed.", "doi": "10.1103/PhysRevLett.57.2935", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1986-12-08", "series_number": "23", "volume": "57", "issue": "23", "pages": "2935-2938" }, { "id": "authors:pk9aa-thb76", "collection": "authors", "collection_id": "pk9aa-thb76", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170811-083138197", "type": "article", "title": "An eight-mode lorenz model of travelling waves in binary fluid convection", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "An eight-mode truncation of a model of binary fluid convection, allowing travelling wave solutions, is introduced. Qualitatively new dynamics is found, in much closer correspondence with recent experiment than a previous five-mode truncation that is restricted to standing waves.", "doi": "10.1016/0375-9601(86)90637-7", "issn": "0375-9601", "publisher": "Elsevier", "publication": "Physics Letters A", "publication_date": "1986-11-24", "series_number": "1", "volume": "119", "issue": "1", "pages": "21-24" }, { "id": "authors:5zqyf-eyb84", "collection": "authors", "collection_id": "5zqyf-eyb84", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra86", "type": "article", "title": "Monte Carlo calculation of quantum tunneling in the dilute instanton limit", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "A new approach for estimating small quantum tunneling rates by Monte Carlo calculation is proposed and demonstrated on a simple one-dimensional model. The application to many-body situations such as atomic exchange in solid 3He is discussed.", "doi": "10.1103/PhysRevA.34.3531", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1986-10-01", "series_number": "4", "volume": "34", "issue": "4", "pages": "3531-3534" }, { "id": "authors:57vb3-dnq64", "collection": "authors", "collection_id": "57vb3-dnq64", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprb86", "type": "article", "title": "Nearest-neighbor exchange in solid 3He", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Bhatt", "given_name": "R. N.", "clpid": "Bhatt-R-N" } ], "abstract": "The effect of including nearest-neighbor exchange into the spin Hamiltonian proposed for solid 3He is studied using the full diagonalization of a 16-quantum-spin cluster.", "doi": "10.1103/PhysRevB.33.7809", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "1986-06-01", "series_number": "11", "volume": "33", "issue": "11", "pages": "7809-7810" }, { "id": "authors:xxt1t-24b52", "collection": "authors", "collection_id": "xxt1t-24b52", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GREpra85", "type": "article", "title": "Stability analysis of two-dimensional models of three-dimensional convection", "author": [ { "family_name": "Greenside", "given_name": "H. S.", "clpid": "Greenside-H-S" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Analytical and numerical methods are used to study the linear stability of spatially periodic solutions for various two-dimensional equations which model thermal convection in fluids. This analysis suggests new model equations that will be useful for investigating questions such as wave-number selection, pattern formation, and the onset of turbulence in large-aspect-ratio Rayleigh-B\u00e9nard systems. In particular, we construct a nonrelaxational model that has stability boundaries similar to those calculated for intermediate Prandtl-number fluids.", "doi": "10.1103/PhysRevA.31.2492", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1985-04-01", "series_number": "4", "volume": "31", "issue": "4", "pages": "2492-2501" }, { "id": "authors:e5sb9-arb38", "collection": "authors", "collection_id": "e5sb9-arb38", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FISprl84", "type": "article", "title": "Comment on \"Dynamics of Charge-Density Waves Pinned by Impurities\"", "author": [ { "family_name": "Fisher", "given_name": "D. S.", "clpid": "Fisher-D-S" }, { "family_name": "Coppersmith", "given_name": "S. N.", "clpid": "Coppersmith-S-N" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "In a recent Letter Klemm and Schrieffer(1) (KS) have attempted to calculate the dynamics of a single domain of a charge-density wave (CDW) pinned to impurities.(2) They give results for the dc current voltage characteristic, and also suggest that the impurity averaged current-current correlation function shows coherent oscillations. Their paper contains several serious errors which invalidate the results even for the very limited case considered of independent domains.", "doi": "10.1103/PhysRevLett.52.481", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1984-02-06", "series_number": "6", "volume": "52", "issue": "6", "pages": "481" }, { "id": "authors:k2w62-g1432", "collection": "authors", "collection_id": "k2w62-g1432", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra84", "type": "article", "title": "Wave-number selection by soft boundaries near threshold", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "An explanation is proposed for the increase in the width of the band of allowed states near threshold found in recent experiments by Cannell, Dominguez-Lerma, and Ahlers on wave-number selection in a Taylor-Couette system. A simple parametrization of the forcing of the motion by the nonuniform gap between the cylinders used to yield the selection produces many features of the data.", "doi": "10.1103/PhysRevA.29.391", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1984-01", "series_number": "1", "volume": "29", "issue": "1", "pages": "391-392" }, { "id": "authors:mns4n-4f713", "collection": "authors", "collection_id": "mns4n-4f713", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BRApra83", "type": "article", "title": "Phase dynamics for the wavy vortex state of the Taylor instability", "author": [ { "family_name": "Brand", "given_name": "H.", "clpid": "Brand-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We present dynamic equations for the slow macroscopic variables of the wavy vortex state. The static solutions explain in a qualitative way recent experimental results of Ahlers et al. on the variation of the vortex diameter throughout the cell. In addition, we predict the existence of a pair of propagating or overdamped normal modes (depending on the wave vector of the disturbance) formed by the slow variables.", "doi": "10.1103/PhysRevA.27.1237", "issn": "0556-2791", "publisher": "American Physical Society", "publication": "Physical Review A", "publication_date": "1983-02", "series_number": "2", "volume": "27", "issue": "2", "pages": "1237-1239" }, { "id": "authors:a4p32-apn12", "collection": "authors", "collection_id": "a4p32-apn12", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra83", "type": "article", "title": "Phase dynamics of convective rolls", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The equation of motion for the slow time dependence of convective rolls due to long-wavelength inhomogeneities is shown to have a singular dependence on the wave vector of the disturbance. Consequences for the skew varicose instability and the wave-number selection principle in textures of curved rolls suggested by Pomeau and Manneville are discussed.", "doi": "10.1103/PhysRevA.27.490", "issn": "1050-2947", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1983-01-01", "series_number": "1", "volume": "27", "issue": "1", "pages": "490-498" }, { "id": "authors:jbg9p-tz036", "collection": "authors", "collection_id": "jbg9p-tz036", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KRAprl82", "type": "article", "title": "Wavelength Selection in Systems Far from Equilibrium", "author": [ { "family_name": "Kramer", "given_name": "Lorenz", "clpid": "Kramer-L" }, { "family_name": "Ben-Jacob", "given_name": "Eshel", "clpid": "Ben-Jacob-E" }, { "family_name": "Brand", "given_name": "Helmut", "clpid": "Brand-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "It is shown that for systems developing stationary periodic patterns there exists at most one stable wavelength state if a supercritical region is connected to a subcritical one by the imposition of a slow spatial variation of the external parameters. In nonpotential systems the selected wavelength depends on the particular combination of parameters that vary but not on the (slow) rate of spatial variation. Suitable parameter variations force the system into a dynamic state.", "doi": "10.1103/PhysRevLett.49.1891", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1982-12-27", "series_number": "26", "volume": "49", "issue": "26", "pages": "1891-1894" }, { "id": "authors:g2gga-1ac06", "collection": "authors", "collection_id": "g2gga-1ac06", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:BRAprl82", "type": "article", "title": "Explanation of flow dissipation in 3He-B", "author": [ { "family_name": "Brand", "given_name": "H.", "clpid": "Brand-H" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We propose an explanation for the large dissipation observed by Eisenstein and Packard(1) in U-tube oscillations of superfluid 3He-B. It turns out that the qualitative change in behavior (compared to superfluid 4He) is simply due to the increased viscosities, which are four orders of magnitude larger in superfluid 3He than in 4He.(2) This means that a dissipation mechanism, not previously discussed (and negligible in 4He), becomes important. The observation does not therefore bring into doubt our basic understanding of the superfluid phases.", "doi": "10.1103/PhysRevLett.49.1959", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1982-12-27", "series_number": "26", "volume": "49", "issue": "26", "pages": "1959" }, { "id": "authors:716yh-qmd03", "collection": "authors", "collection_id": "716yh-qmd03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SNEprl82", "type": "article", "title": "Sliding conductivity of charge-density waves", "author": [ { "family_name": "Sneddon", "given_name": "Leigh", "clpid": "Sneddon-L" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Fisher", "given_name": "Daniel S.", "clpid": "Fisher-D-S" } ], "abstract": "A classical model of charge-density wave motion at large velocities is used to describe the nonlinear response of the charge-density wave in fields well above the threshold. The charge-density wave is regarded as a charged, deformable medium and its interaction with pinning centers is treated in perturbation theory. The results fit the available high-field data on NbSe3 well. The observed interference effects between a large dc field and an ac field are also explained.", "doi": "10.1103/PhysRevLett.49.292", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1982-07-26", "series_number": "4", "volume": "49", "issue": "4", "pages": "292-295" }, { "id": "authors:3yb9t-f2m24", "collection": "authors", "collection_id": "3yb9t-f2m24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpof82", "type": "article", "title": "Boundary conditions on the envelope function of convective rolls close to onset", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "Boundary conditions are derived for the envelope function of convective rolls approaching a rigid sidewall at an arbitrary orientation. This generalizes previous results for parallel and perpendicular rolls. It provides the first step in a study of the convective pattern to be expected close to onset in Rayleigh\u2013Benard cells of any cross section, and allows consideration of more complicated patterns in rectangular or circular cross sections.", "doi": "10.1063/1.863835", "issn": "0031-9171", "publisher": "Physics of Fluids", "publication": "Physics of Fluids", "publication_date": "1982-06-01", "series_number": "6", "volume": "25", "issue": "6", "pages": "936-941" }, { "id": "authors:mbgd0-hp485", "collection": "authors", "collection_id": "mbgd0-hp485", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra82", "type": "article", "title": "Ingredients of a theory of convective textures close to onset", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The patterns of convection to be expected in laterally large Rayleigh-B\u00e9nard cells close to onset are analyzed using the amplitude equation approach of Newell and Whitehead and Segel. This equation allows the introduction of a Lyapunov functional, which may be used to order the stability of various patterns. Two competing effects are identified: a surface effect that favors rolls approaching the sidewalls normally, and so tends to produce O(1) rotations of the roll orientation over the cell, and bulk effects that favor straight parallel rolls. The competition between these effects, and the role of defects in the structure, are studied in some simple examples.", "doi": "10.1103/PhysRevA.25.1065", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1982-02-01", "series_number": "2", "volume": "25", "issue": "2", "pages": "1065-1076" }, { "id": "authors:z1q0v-9wp55", "collection": "authors", "collection_id": "z1q0v-9wp55", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprl80", "type": "article", "title": "Effect of Distant Sidewalls on Wave-Number Selection in Rayleigh-B\u00e9nard Convection", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Daniels", "given_name": "P. G.", "clpid": "Daniels-P-G" }, { "family_name": "Hohenberg", "given_name": "P. C.", "clpid": "Hohenberg-P-C" }, { "family_name": "Siggia", "given_name": "E. D.", "clpid": "Siggia-E-D" } ], "abstract": "An analysis is presented of the steady states of two-dimensional convection in a laterally finite rectangular container near threshold. It is shown that the presence of sidewalls severely restricts the allowed wave vectors which can occur in the bulk of the container. This effect provides a possible mechanism to explain the observed wavelength increase of convective rolls with increasing Rayleigh numbers.", "doi": "10.1103/PhysRevLett.45.898", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1980-09-15", "series_number": "11", "volume": "45", "issue": "11", "pages": "898-901" }, { "id": "authors:jqeb1-1jc18", "collection": "authors", "collection_id": "jqeb1-1jc18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpof80", "type": "article", "title": "Derivation of the amplitude equation at the Rayleigh\u2013B\u00e8nard instability", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The full amplitude equation (as introduced by Newell and Whitehead and Segel) is derived directly from the hydrodynamic equations for both \"free\" and \"rigid\" upper and lower boundaries. The coefficients involved, including the interaction parameter of nonparallel rolls, are explicitly calculated for both boundary conditions and all Prandtl numbers.", "doi": "10.1063/1.863198", "issn": "0031-9171", "publisher": "Physics of Fluids", "publication": "Physics of Fluids", "publication_date": "1980-09-01", "series_number": "9", "volume": "23", "issue": "9", "pages": "1727-1731" }, { "id": "authors:ac59q-vwm18", "collection": "authors", "collection_id": "ac59q-vwm18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:STAprl80", "type": "article", "title": "X-Ray Diffraction Intensities of a Smectic-A Liquid Crystal", "author": [ { "family_name": "Stamatoff", "given_name": "J.", "clpid": "Stamatoff-J" }, { "family_name": "Cladis", "given_name": "P. E.", "clpid": "Cladis-P-E" }, { "family_name": "Guillon", "given_name": "D.", "clpid": "Guillon-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Bilash", "given_name": "T.", "clpid": "Bilash-T" }, { "family_name": "Finn", "given_name": "P.", "clpid": "Finn-P" } ], "abstract": "Higher-order diffraction from the smectic-A liquid-crystalline phase of cyanooctylbiphenyl (COB) has been measured. The dominant short-range disorder, described by the relative intensities, is in agreement with our estimate of the smectic order parameter based on McMillan's mean-field theory.", "doi": "10.1103/PhysRevLett.44.1509", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1980-06-09", "series_number": "23", "volume": "44", "issue": "23", "pages": "1509-1512" }, { "id": "authors:v3xtp-a2n02", "collection": "authors", "collection_id": "v3xtp-a2n02", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:OSHprl80", "type": "article", "title": "Nuclear Antiferromagnetic Resonance in Solid 3He", "author": [ { "family_name": "Osheroff", "given_name": "D. D.", "clpid": "Osheroff-D-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Fisher", "given_name": "D. S.", "clpid": "Fisher-D-S" } ], "abstract": "Detailed measurements of the low-field antiferromagnetic resonance spectrum of spin-ordered bcc 3He exhibit large shifts from the Larmor frequency, with a zero-field resonant frequency near zero temperature of \u03a90/2\u03c0\u2243825 kHz. Analysis of the spectrum leads to stringent constraints on possible sublattice structures. The temperature dependence of \u03a90 shows low-temperature behavior expected from spin-wave theory, and indicates a first-order transition at 1.03 mK.", "doi": "10.1103/PhysRevLett.44.792", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1980-03-24", "series_number": "12", "volume": "44", "issue": "12", "pages": "792-795" }, { "id": "authors:x3hgd-bgb57", "collection": "authors", "collection_id": "x3hgd-bgb57", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprb79b", "type": "article", "title": "Effect of magnetic fields on a spin-Peierls transition", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The effect of a magnetic field on the transition temperature of a spin-Peierls transition and the wave vector of the lattice distortion is calculated using the theory based on the Luther-Peschel-type treatment of the spin-correlation functions. Comparison with experiment offers the prospect of testing the predictions of this theory. A brief discussion of the nature of the commensurate to incommensurate transition is included.", "doi": "10.1103/PhysRevB.20.4606", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "1979-12-01", "series_number": "11", "volume": "20", "issue": "11", "pages": "4606-4611" }, { "id": "authors:fm8fk-yfx14", "collection": "authors", "collection_id": "fm8fk-yfx14", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:RICprb79", "type": "article", "title": "Dynamics of charge-density waves in the presence of free carriers", "author": [ { "family_name": "Rice", "given_name": "T. M.", "clpid": "Rice-T-M" }, { "family_name": "Lee", "given_name": "P. A.", "clpid": "Lee-P-A" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "The response of an unpinned, incommensurate charge-density wave (CDW) to external perturbations in the presence of free quasiparticles is studied. The CDW can act as a charged lattice but the proportion of the total charge density assigned to the CDW and to quasiparticles depends on the external conditions. If the collisions between quasiparticles can be ignored, the effective charge of the CDW is different in the static and dynamic limits. In the static limit the quasiparticles relax to screen spatial variations and the form is analogous to that used for superfluids. In the dynamic limit the quasiparticles do not renormalize the effective charge. In the presence of collisions the effective charge depends on the distribution of quasiparticles which in turn depends on the relative importance of various scattering rates. In general, a simple two-fluid model is not applicable.", "doi": "10.1103/PhysRevB.20.1345", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "1979-08-15", "series_number": "4", "volume": "20", "issue": "4", "pages": "1345-1352" }, { "id": "authors:f4rbx-w4281", "collection": "authors", "collection_id": "f4rbx-w4281", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:LIUprl79", "type": "article", "title": "Gauge Wheel of Superfluid ^3He", "author": [ { "family_name": "Liu", "given_name": "Mario", "clpid": "Liu-Mario" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "A change in the phase of the order parameter of ^3He-A can be undone by a subsequent rotation. We investigate the dynamical consequence of this broken relative gauge and rotational symmetry. In particular a wheel rotated in the liquid acts as a \"gauge transformer\" driving a superflow. Such experiments provide a very direct probe of this unusual feature of the order parameter, at the same time measuring the orbital quantum number of the pairs.", "doi": "10.1103/PhysRevLett.43.296", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1979-07-23", "series_number": "4", "volume": "43", "issue": "4", "pages": "296-299" }, { "id": "authors:s9vz1-mjd18", "collection": "authors", "collection_id": "s9vz1-mjd18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:STEprl79", "type": "article", "title": "Phase Transitions in Two-Dimensional Superfluid 3He", "author": [ { "family_name": "Stein", "given_name": "D. L.", "clpid": "Stein-D-L" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "A discussion of phase transitions in two-dimensional 3He is presented. A new type of phase transition is proposed, in which \"islands of reversed l\u2192\" spontaneously nucleate, leading to an Ising-like transition. It is shown that such transitions may need to be taken into account, possibly competing with the well-know Kosterlitz-Thouless transition. Consequences of both types of transition are discussed.", "doi": "10.1103/PhysRevLett.42.504", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1979-02-19", "series_number": "8", "volume": "42", "issue": "8", "pages": "504-507" }, { "id": "authors:gy05k-zva66", "collection": "authors", "collection_id": "gy05k-zva66", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprb79a", "type": "article", "title": "A new theory of the spin-Peierls transition with special relevance to the experiments on TTFCuBDT", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Fisher", "given_name": "Daniel S.", "clpid": "Fisher-D-S" } ], "abstract": "We develop a new theory of the spin-Peierls transition in spin-\u00bd Heisenberg chains, treating the phonons in a mean-field random-phase approximation (RPA) as in previous work, but calculating the relevant response functions of the spin system using the procedure of Luther and Peschel. We show that the RPA on the phonons (and therefore our whole calculation) should be good for the experimentally important system tetrathiafulvalenium bis-cis- (1,2-perfluoromethylethylene-1,2-dithiolato)-copper (TTFCuBDT). It is also exact for a model system in which planes of atoms perpendicular to the chains are constrained by lattice forces to move together: we have derived some exact results for the spin-Peierls transition in this model system. We find a new linear dependence of the transition temperature Tc on the spin-phonon coupling, and an enhancement of Tc and also of the rate of phonon softening above Tc. Predictions of some other signatures of the transition, such as the specific-heat jump ratio, are however not much changed from previous work. Exact results are found for the leading dependence of the ground-state energy E and gap \u0394 in the excitation spectrum on the lattice distortion \u03b4: E\u221d\u03b4^(4/3), \u0394\u221d\u03b4^(2/3).", "doi": "10.1103/PhysRevB.19.402", "issn": "0163-1829", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "1979-01-01", "series_number": "1", "volume": "19", "issue": "1", "pages": "402-419" }, { "id": "authors:64xm0-qxd52", "collection": "authors", "collection_id": "64xm0-qxd52", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:LIUprl78", "type": "article", "title": "Broken Spin-Orbit Symmetry in Superfluid ^3He and B-Phase Dynamics", "author": [ { "family_name": "Liu", "given_name": "Mario", "clpid": "Liu-Mario" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "It is shown that the B phase not only has a \"phase dynamics\" such as that in He II, but also an orbital dynamics somewhat similar to that of 3He-A or a nematic liquid cyrstal. The rederived B-phase hydrodynamic theory, complete with nonlinear terms, establishes the equivalence between rotations in orbital and spin spaces and shows the intriguing possibilities of generating a magnetization by mechanical rotations and of substituting the magnetic field with oscillating parallel plates in \"NMR\" experiments.", "doi": "10.1103/PhysRevLett.41.250", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1978-07-24", "series_number": "4", "volume": "41", "issue": "4", "pages": "250-253" }, { "id": "authors:g9wcn-b1h56", "collection": "authors", "collection_id": "g9wcn-b1h56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROjpc78", "type": "article", "title": "Stability of the aligned state of ^3He-A in a superflow", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" }, { "family_name": "Liu", "given_name": "M.", "clpid": "Liu-Mario" } ], "abstract": "The stability of the equilibrium orientation of l parallel or antiparallel to a counterflow is studied both dynamically and statically. In contrast to the recent work of Hall and Hook (1977) - who, it is shown, use incorrect dynamical equations - the equilibrium is found to be stable in the Ginzburg-Landau regime. At lower temperatures an instability should develop.", "doi": "10.1088/0022-3719/11/9/018", "issn": "0022-3719", "publisher": "IOP", "publication": "Journal of Physics C: Solid State Physics", "publication_date": "1978-05-14", "series_number": "9", "volume": "11", "issue": "9", "pages": "1795-1800" }, { "id": "authors:6ag3s-vr519", "collection": "authors", "collection_id": "6ag3s-vr519", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:OSHprl77", "type": "article", "title": "Interfacial Surface Energy between the Superfluid Phases of He3", "author": [ { "family_name": "Osheroff", "given_name": "D. D.", "clpid": "Osheroff-D-D" }, { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "We report the first measurements of \u03c3AB, the surface energy associated with the interface between the A and B phases of superfluid 3He as they coexist at TAB. At melting pressure we find \u03c3AB=6\u00d710^-6 erg cm^-2 in no magnetic field (TAB/TC=0.79) rising to \u03c3AB=1.6\u00d710^-5 erg cm^-2 in a magnetic field of 4 kOe (TAB/TC=0.5). Theoretical calculations that we present give an estimate about 50% larger.", "doi": "10.1103/PhysRevLett.38.905", "issn": "0031-9007", "publisher": "American Physical Society", "publication": "Physical Review Letters", "publication_date": "1977-04-18", "series_number": "16", "volume": "38", "issue": "16", "pages": "905-909" }, { "id": "authors:sx1z6-n3924", "collection": "authors", "collection_id": "sx1z6-n3924", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROprb77", "type": "article", "title": "Charge states of fast protons in solids", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "It is shown that charge states of high-energy protons in solids may be simply understood in terms of capture into, and subsequent loss from, bound states on the proton within the solid. Both the equilibrium charge states emerging from the solid, and the approach to equilibrium within the solid, may be calculated. At high enough proton velocities gas cross sections may be used to predict the results in solids. This is directly confirmed by experimental results, taken from the literature, on solid carbon and gaseous carbon compounds.", "doi": "10.1103/PhysRevB.15.602", "issn": "0556-2805", "publisher": "Physical Review B", "publication": "Physical Review B", "publication_date": "1977-01-15", "series_number": "2", "volume": "15", "issue": "2", "pages": "602-607" }, { "id": "authors:wef1a-a6p23", "collection": "authors", "collection_id": "wef1a-a6p23", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:CROpra74", "type": "article", "title": "Impulse given to a plate by a quantized vortex ring", "author": [ { "family_name": "Cross", "given_name": "M. C.", "clpid": "Cross-M-C" } ], "abstract": "It is shown that a quantized vortex ring in superfluid He4, moving freely towards a flat plate, gives to it an impulse equal to the fluid impulse of the ring, provided only that the radius of the plate is larger than the initial vortex radius but smaller than the distance to the source of the ring. This is in agreement with experiment, in contradiction to two previous theoretical analyses.", "doi": "10.1103/PhysRevA.10.1442", "issn": "0556-2791", "publisher": "Physical Review A", "publication": "Physical Review A", "publication_date": "1974-10-01", "series_number": "4", "volume": "10", "issue": "4", "pages": "1442-1445" } ]