1
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Ganapa S. Quasiperiodicity in the α-Fermi-Pasta-Ulam-Tsingou problem revisited: An approach using ideas from wave turbulence. CHAOS (WOODBURY, N.Y.) 2023; 33:093102. [PMID: 37656916 DOI: 10.1063/5.0154157] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Accepted: 08/14/2023] [Indexed: 09/03/2023]
Abstract
The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work, we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α-FPUT system in more detail. We aim to reconstruct the quasiperiodic behavior observed in the original paper from the canonical transformation used to remove the three-wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work, we find that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three-wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α-FPUT system and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α-FPUT problem. We argue that certain aspects of the α-FPUT system such as thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.
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2
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Li Z, Porter MA, Choubey B. Recurrence recovery in heterogeneous Fermi-Pasta-Ulam-Tsingou systems. CHAOS (WOODBURY, N.Y.) 2023; 33:093108. [PMID: 37676112 DOI: 10.1063/5.0154970] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2023] [Accepted: 08/04/2023] [Indexed: 09/08/2023]
Abstract
The computational investigation of Fermi, Pasta, Ulam, and Tsingou (FPUT) of arrays of nonlinearly coupled oscillators has led to a wealth of studies in nonlinear dynamics. Most studies of oscillator arrays have considered homogeneous oscillators, even though there are inherent heterogeneities between individual oscillators in real-world arrays. Well-known FPUT phenomena, such as energy recurrence, can break down in such heterogeneous systems. In this paper, we present an approach-the use of structured heterogeneities-to recover recurrence in FPUT systems in the presence of oscillator heterogeneities. We examine oscillator variabilities in FPUT systems with cubic nonlinearities, and we demonstrate that centrosymmetry in oscillator arrays may be an important source of recurrence.
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Affiliation(s)
- Zidu Li
- Department of Electrical Engineering and Computer Science, University of Siegen, Siegen, North Rhine-Westphalia 57072, Germany
| | - Mason A Porter
- Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095, USA
- Santa Fe Institute, Santa Fe, New Mexico 87501, USA
| | - Bhaskar Choubey
- Department of Electrical Engineering and Computer Science, University of Siegen, Siegen, North Rhine-Westphalia 57072, Germany
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3
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Reiss KA, Campbell DK. The Metastable State of Fermi-Pasta-Ulam-Tsingou Models. ENTROPY (BASEL, SWITZERLAND) 2023; 25:300. [PMID: 36832666 PMCID: PMC9954834 DOI: 10.3390/e25020300] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/19/2022] [Revised: 01/30/2023] [Accepted: 01/30/2023] [Indexed: 06/18/2023]
Abstract
Classical statistical mechanics has long relied on assumptions such as the equipartition theorem to understand the behavior of the complicated systems of many particles. The successes of this approach are well known, but there are also many well-known issues with classical theories. For some of these, the introduction of quantum mechanics is necessary, e.g., the ultraviolet catastrophe. However, more recently, the validity of assumptions such as the equipartition of energy in classical systems was called into question. For instance, a detailed analysis of a simplified model for blackbody radiation was apparently able to deduce the Stefan-Boltzmann law using purely classical statistical mechanics. This novel approach involved a careful analysis of a "metastable" state which greatly delays the approach to equilibrium. In this paper, we perform a broad analysis of such a metastable state in the classical Fermi-Pasta-Ulam-Tsingou (FPUT) models. We treat both the α-FPUT and β-FPUT models, exploring both quantitative and qualitative behavior. After introducing the models, we validate our methodology by reproducing the well-known FPUT recurrences in both models and confirming earlier results on how the strength of the recurrences depends on a single system parameter. We establish that the metastable state in the FPUT models can be defined by using a single degree-of-freedom measure-the spectral entropy (η)-and show that this measure has the power to quantify the distance from equipartition. For the α-FPUT model, a comparison to the integrable Toda lattice allows us to define rather clearly the lifetime of the metastable state for the standard initial conditions. We next devise a method to measure the lifetime of the metastable state tm in the α-FPUT model that reduces the sensitivity to the exact initial conditions. Our procedure involves averaging over random initial phases in the plane of initial conditions, the P1-Q1 plane. Applying this procedure gives us a power-law scaling for tm, with the important result that the power laws for different system sizes collapse down to the same exponent as Eα2→0. We examine the energy spectrum E(k) over time in the α-FPUT model and again compare the results to those of the Toda model. This analysis tentatively supports a method for an irreversible energy dissipation process suggested by Onorato et al.: four-wave and six-wave resonances as described by the "wave turbulence" theory. We next apply a similar approach to the β-FPUT model. Here, we explore in particular the different behavior for the two different signs of β. Finally, we describe a procedure for calculating tm in the β-FPUT model, a very different task than for the α-FPUT model, because the β-FPUT model is not a truncation of an integrable nonlinear model.
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4
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Wu X, Zhang Y, Peng J, Boscolo S, Finot C, Zeng H. Farey tree and devil's staircase of frequency-locked breathers in ultrafast lasers. Nat Commun 2022; 13:5784. [PMID: 36184670 PMCID: PMC9527256 DOI: 10.1038/s41467-022-33525-0] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2021] [Accepted: 09/21/2022] [Indexed: 02/08/2023] Open
Abstract
Nonlinear systems with two competing frequencies show locking or resonances. In lasers, the two interacting frequencies can be the cavity repetition rate and a frequency externally applied to the system. Conversely, the excitation of breather oscillations in lasers naturally triggers a second characteristic frequency in the system, therefore showing competition between the cavity repetition rate and the breathing frequency. Yet, the link between breathing solitons and frequency locking is missing. Here we demonstrate frequency locking at Farey fractions of a breather laser. The winding numbers exhibit the hierarchy of the Farey tree and the structure of a devil’s staircase. Numerical simulations of a discrete laser model confirm the experimental findings. The breather laser may therefore serve as a simple test bed to explore ubiquitous synchronization dynamics of nonlinear systems. The locked breathing frequencies feature a high signal-to-noise ratio and can give rise to dense radio-frequency combs, which are attractive for applications. Fractal optical solitons were studied in theory while it is cumbersome their experimental realization in optics setups. Here, the authors find that breathing solitons in lasers constitute fractals―the devil’s staircases, which are around 3000 times more stable than classical ones.
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Affiliation(s)
- Xiuqi Wu
- State Key Laboratory of Precision Spectroscopy, East China Normal University, 200241, Shanghai, China
| | - Ying Zhang
- State Key Laboratory of Precision Spectroscopy, East China Normal University, 200241, Shanghai, China
| | - Junsong Peng
- State Key Laboratory of Precision Spectroscopy, East China Normal University, 200241, Shanghai, China. .,Collaborative Innovation Center of Extreme Optics, Shanxi University, 030006, Taiyuan, Shanxi, China.
| | - Sonia Boscolo
- Aston Institute of Photonic Technologies, Aston University, Birmingham, B4 7ET, UK
| | - Christophe Finot
- Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS-Université de Bourgogne Franche-Comté, F-21078, Dijon, Cedex, France
| | - Heping Zeng
- State Key Laboratory of Precision Spectroscopy, East China Normal University, 200241, Shanghai, China. .,Chongqing Key Laboratory of Precision Optics, Chongqing Institute of East China Normal University, 401120, Chongqing, China. .,Shanghai Research Center for Quantum Sciences, 201315, Shanghai, China.
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5
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Iñarrea M, González-Férez R, Salas JP, Schmelcher P. Chaos and thermalization in a classical chain of dipoles. Phys Rev E 2022; 106:014213. [PMID: 35974544 DOI: 10.1103/physreve.106.014213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/20/2022] [Accepted: 05/19/2022] [Indexed: 06/15/2023]
Abstract
We explore the connection between chaos, thermalization, and ergodicity in a linear chain of N interacting dipoles. Starting from the ground state, and considering chains of different numbers of dipoles, we introduce single site excitations with excess energy ΔK. The time evolution of the chaoticity of the system and the energy localization along the chain is analyzed by computing, up to a very long time, the statistical average of the finite-time Lyapunov exponent λ(t) and the participation ratio Π(t). For small ΔK, the evolution of λ(t) and Π(t) indicates that the system becomes chaotic at approximately the same time as Π(t) reaches a steady state. For the largest considered values of ΔK the system becomes chaotic at an extremely early stage in comparison with the energy relaxation times. We find that this fact is due to the presence of chaotic breathers that keep the system far from equipartition and ergodicity. Finally, we show numerically and analytically that the asymptotic values attained by the participation ratio Π(t) fairly correspond to thermal equilibrium.
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Affiliation(s)
- Manuel Iñarrea
- Área de Física, Universidad de La Rioja, 26006 Logroño, La Rioja, Spain
| | - Rosario González-Férez
- Instituto Carlos I de Física Teórica y Computacional, and Departamento de Física Atómica, Molecular y Nuclear, Universidad de Granada, 18071 Granada, Spain
| | - J Pablo Salas
- Área de Física, Universidad de La Rioja, 26006 Logroño, La Rioja, Spain
| | - Peter Schmelcher
- The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
- Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
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Mithun T, Danieli C, Fistul MV, Altshuler BL, Flach S. Fragile many-body ergodicity from action diffusion. Phys Rev E 2021; 104:014218. [PMID: 34412341 DOI: 10.1103/physreve.104.014218] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/12/2020] [Accepted: 07/01/2021] [Indexed: 11/07/2022]
Abstract
Weakly nonintegrable many-body systems can restore ergodicity in distinctive ways depending on the range of the interaction network in action space. Action resonances seed chaotic dynamics into the networks. Long-range networks provide well connected resonances with ergodization controlled by the individual resonance chaos time scales. Short-range networks instead yield a dramatic slowing down of ergodization in action space, and lead to rare resonance diffusion. We use Josephson junction chains as a paradigmatic study case. We exploit finite time average distributions to characterize the thermalizing dynamics of actions. We identify an action resonance diffusion regime responsible for the slowing down. We extract the diffusion coefficient of that slow process and measure its dependence on the proximity to the integrable limit. Independent measures of correlation functions confirm our findings. The observed fragile diffusion is relying on weakly chaotic dynamics in spatially isolated action resonances. It can be suppressed, and ergodization delayed, by adding weak action noise, as a proof of concept.
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Affiliation(s)
- Thudiyangal Mithun
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.,Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
| | - Carlo Danieli
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea.,Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | - M V Fistul
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea.,Theoretische Physik III, Ruhr-Universität Bochum, Bochum 44801, Germany.,Russian Quantum Center, National University of Science and Technology "MISIS", 119049 Moscow, Russia
| | - B L Altshuler
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea.,Physics Department, Columbia University, New York, New York 10027, USA
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea.,Basic Science Program (IBS School), Korea University of Science and Technology (UST), Daejeon 34113, Korea
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7
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Yin HM, Pan Q, Chow KW. Four-wave mixing and coherently coupled Schrödinger equations: Cascading processes and Fermi-Pasta-Ulam-Tsingou recurrence. CHAOS (WOODBURY, N.Y.) 2021; 31:083117. [PMID: 34470240 DOI: 10.1063/5.0051584] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/26/2021] [Accepted: 07/21/2021] [Indexed: 06/13/2023]
Abstract
Modulation instability, breather formation, and the Fermi-Pasta-Ulam-Tsingou recurrence (FPUT) phenomena are studied in this article. Physically, such nonlinear systems arise when the medium is slightly anisotropic, e.g., optical fibers with weak birefringence where the slowly varying pulse envelopes are governed by these coherently coupled Schrödinger equations. The Darboux transformation is used to calculate a class of breathers where the carrier envelope depends on the transverse coordinate of the Schrödinger equations. A "cascading mechanism" is utilized to elucidate the initial stages of FPUT. More precisely, higher order nonlinear terms that are exponentially small initially can grow rapidly. A breather is formed when the linear mode and higher order ones attain roughly the same magnitude. The conditions for generating various breathers and connections with modulation instability are elucidated. The growth phase then subsides and the cycle is repeated, leading to FPUT. Unequal initial conditions for the two waveguides produce symmetry breaking, with "eye-shaped" breathers in one waveguide and "four-petal" modes in the other. An analytical formula for the time or distance of breather formation for a two-waveguide system is proposed, based on the disturbance amplitude and instability growth rate. Excellent agreement with numerical simulations is achieved. Furthermore, the roles of modulation instability for FPUT are elucidated with illustrative case studies. In particular, depending on whether the second harmonic falls within the unstable band, FPUT patterns with one single or two distinct wavelength(s) are observed. For applications to temporal optical waveguides, the present formulation can predict the distance along a weakly birefringent fiber needed to observe FPUT.
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Affiliation(s)
- H M Yin
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong
| | - Q Pan
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong
| | - K W Chow
- Department of Mechanical Engineering, University of Hong Kong, Pokfulam, Hong Kong
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8
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Pace SD, Reiss KA, Campbell DK. The β Fermi-Pasta-Ulam-Tsingou recurrence problem. CHAOS (WOODBURY, N.Y.) 2019; 29:113107. [PMID: 31779356 DOI: 10.1063/1.5122972] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2019] [Accepted: 10/21/2019] [Indexed: 06/10/2023]
Abstract
We perform a thorough investigation of the first Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in the β-FPUT chain for both positive and negative β. We show numerically that the rescaled FPUT recurrence time Tr=tr/(N+1)3 depends, for large N, only on the parameter S≡Eβ(N+1). Our numerics also reveal that for small |S|, Tr is linear in S with positive slope for both positive and negative β. For large |S|, Tr is proportional to |S|-1/2 for both positive and negative β but with different multiplicative constants. We numerically study the continuum limit and find that the recurrence time closely follows the |S|-1/2 scaling and can be interpreted in terms of solitons, as in the case of the KdV equation for the α chain. The difference in the multiplicative factors between positive and negative β arises from soliton-kink interactions that exist only in the negative β case. We complement our numerical results with analytical considerations in the nearly linear regime (small |S|) and in the highly nonlinear regime (large |S|). For the former, we extend previous results using a shifted-frequency perturbation theory and find a closed form for Tr that depends only on S. In the latter regime, we show that Tr∝|S|-1/2 is predicted by the soliton theory in the continuum limit. We then investigate the existence of the FPUT recurrences and show that their disappearance surprisingly depends only on Eβ for large N, not S. Finally, we end by discussing the striking differences in the amount of energy mixing between positive and negative β and offer some remarks on the thermodynamic limit.
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Affiliation(s)
- Salvatore D Pace
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Kevin A Reiss
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - David K Campbell
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
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9
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Danieli C, Mithun T, Kati Y, Campbell DK, Flach S. Dynamical glass in weakly nonintegrable Klein-Gordon chains. Phys Rev E 2019; 100:032217. [PMID: 31639954 DOI: 10.1103/physreve.100.032217] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/27/2018] [Indexed: 06/10/2023]
Abstract
Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale T_{E} on which these distributions converge to δ distributions. We relate T_{E} to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations σ_{τ}^{+} dominating the means μ_{τ}^{+} and establish that T_{E}∼(σ_{τ}^{+})^{2}/μ_{τ}^{+}. The Lyapunov time T_{Λ} (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks T_{Λ}≈σ_{τ}^{+}, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a dynamical glass, where T_{E} grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies T_{Λ}≲μ_{τ}^{+}. This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time T_{E}.
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Affiliation(s)
- Carlo Danieli
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
| | - Thudiyangal Mithun
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
| | - Yagmur Kati
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
- Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Republic of Korea
| | - David K Campbell
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Republic of Korea
- New Zealand Institute for Advanced Study, Massey University, Auckland 02215, New Zealand
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10
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Goldfriend T, Kurchan J. Equilibration of quasi-integrable systems. Phys Rev E 2019; 99:022146. [PMID: 30934235 DOI: 10.1103/physreve.99.022146] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2018] [Indexed: 06/09/2023]
Abstract
We study the slow relaxation of isolated quasi-integrable systems, focusing on the classical problem of Fermi-Pasta-Ulam-Tsingou (FPU) chain. It is well known that the initial energy sharing between different linear modes can be inferred by the integrable Toda chain. Using numerical simulations, we show explicitly how the relaxation of the FPU chain toward equilibration is determined by a slow drift within the space of Toda's integrals of motion. We analyze the whole spectrum of Toda modes and show how they dictate, via a generalized Gibbs ensemble, the quasistatic states along the FPU evolution. This picture is employed to devise a fast numerical integration, which can be generalized to other quasi-integrable models. In addition, we discuss how a fluctuation theorem, recently derived in Goldfriend and Kurchan [Europhys. Lett. 124, 10002 (2018)10.1209/0295-5075/124/10002], describes the large deviations as the system flows in the entropy landscape.
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Affiliation(s)
- Tomer Goldfriend
- Laboratoire de Physique Statistique, Département de physique de l'ENS, École Normale Supérieure, PSL Research University 24 rue Lhomond, 75005 Paris, France and Université Paris Diderot, Sorbonne Paris-Cité; Sorbonne Universités, UPMC Univ. Paris 06, CNRS; 24 rue Lhomond, 75005 Paris, France
| | - Jorge Kurchan
- Laboratoire de Physique Statistique, Département de physique de l'ENS, École Normale Supérieure, PSL Research University 24 rue Lhomond, 75005 Paris, France and Université Paris Diderot, Sorbonne Paris-Cité; Sorbonne Universités, UPMC Univ. Paris 06, CNRS; 24 rue Lhomond, 75005 Paris, France
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11
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Barnard AW, Zhang M, Wiederhecker GS, Lipson M, McEuen PL. Real-time vibrations of a carbon nanotube. Nature 2019; 566:89-93. [PMID: 30664747 DOI: 10.1038/s41586-018-0861-0] [Citation(s) in RCA: 27] [Impact Index Per Article: 5.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/05/2017] [Accepted: 11/14/2018] [Indexed: 11/09/2022]
Abstract
The field of miniature mechanical oscillators is rapidly evolving, with emerging applications including signal processing, biological detection1 and fundamental tests of quantum mechanics2. As the dimensions of a mechanical oscillator shrink to the molecular scale, such as in a carbon nanotube resonator3-7, their vibrations become increasingly coupled and strongly interacting8,9 until even weak thermal fluctuations could make the oscillator nonlinear10-13. The mechanics at this scale possesses rich dynamics, unexplored because an efficient way of detecting the motion in real time is lacking. Here we directly measure the thermal vibrations of a carbon nanotube in real time using a high-finesse micrometre-scale silicon nitride optical cavity as a sensitive photonic microscope. With the high displacement sensitivity of 700 fm Hz-1/2 and the fine time resolution of this technique, we were able to discover a realm of dynamics undetected by previous time-averaged measurements and a room-temperature coherence that is nearly three orders of magnitude longer than previously reported. We find that the discrepancy in the coherence stems from long-time non-equilibrium dynamics, analogous to the Fermi-Pasta-Ulam-Tsingou recurrence seen in nonlinear systems14. Our data unveil the emergence of a weakly chaotic mechanical breather15, in which vibrational energy is recurrently shared among several resonance modes-dynamics that we are able to reproduce using a simple numerical model. These experiments open up the study of nonlinear mechanical systems in the Brownian limit (that is, when a system is driven solely by thermal fluctuations) and present an integrated, sensitive, high-bandwidth nanophotonic interface for carbon nanotube resonators.
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Affiliation(s)
- Arthur W Barnard
- School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA.,Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA.,Physics Department, Stanford University, Stanford, CA, USA
| | - Mian Zhang
- School of Applied and Engineering Physics, Cornell University, Ithaca, NY, USA.,School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA.,John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA
| | - Gustavo S Wiederhecker
- School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA.,Gleb Wataghin Physics Institute, University of Campinas, Campinas, Brazil
| | - Michal Lipson
- School of Electrical and Computer Engineering, Cornell University, Ithaca, NY, USA. .,Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, NY, USA. .,Electrical Engineering, Columbia University, New York, NY, USA.
| | - Paul L McEuen
- Laboratory of Atomic and Solid State Physics, Cornell University, Ithaca, NY, USA. .,Kavli Institute at Cornell for Nanoscale Science, Cornell University, Ithaca, NY, USA.
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12
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Zampetaki A, Salas JP, Schmelcher P. Energy transfer mechanisms in a dipole chain: From energy equipartition to the formation of breathers. Phys Rev E 2018; 98:022202. [PMID: 30253512 DOI: 10.1103/physreve.98.022202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/14/2018] [Indexed: 11/07/2022]
Abstract
We study the energy transfer in a classical dipole chain of N interacting rigid rotating dipoles. The underlying high-dimensional potential energy landscape is analyzed in particular by determining the equilibrium points and their stability in the common plane of rotation. Starting from the minimal energy configuration, the response of the chain to excitation of a single dipole is investigated. Using both the linearized and the exact Hamiltonian of the dipole chain, we detect an approximate excitation energy threshold between a weakly and a strongly nonlinear dynamics. In the weakly nonlinear regime, the chain approaches in the course of time the expected energy equipartition among the dipoles. For excitations of higher energy, strongly localized excitations appear whose trajectories in time are either periodic or irregular, relating to the well-known discrete or chaotic breathers, respectively. The phenomenon of spontaneous formation of domains of opposite polarization and phase locking is found to commonly accompany the time evolution of the chaotic breathers. Finally, the sensitivity of the dipole chain dynamics to the initial conditions is studied as a function of the initial excitation energy by computing a fast chaos indicator. The results of this study confirm the aforementioned approximate threshold value for the initial excitation energy, below which the dynamics of the dipole chain is regular and above which it is chaotic.
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Affiliation(s)
- Alexandra Zampetaki
- Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
| | - J Pablo Salas
- Área de Física, Universidad de La Rioja, 26006 Logroño, La Rioja, Spain
| | - Peter Schmelcher
- Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany.,The Hamburg Center for Ultrafast Imaging, Luruper Chaussee 149, 22761 Hamburg, Germany
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13
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Wang Y, Zhu Z, Zhang Y, Huang L. Metastable states and energy flow pathway in square graphene resonators. Phys Rev E 2018; 97:012143. [PMID: 29448334 DOI: 10.1103/physreve.97.012143] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/29/2017] [Indexed: 11/07/2022]
Abstract
Nonlinear interaction between flexural modes is critical to heat conductivity and mechanical vibration of two-dimensional materials such as graphene. Much effort has been devoted to understand the underlying mechanism. In this paper, we examine solely the out-of-plane flexural modes and identify their energy flow pathway during thermalization process. The key is the development of a universal scheme that numerically characterizes the strength of nonlinear interactions between normal modes. In particular, for our square graphene system, the modes are grouped into four classes by their distinct symmetries. The couplings are significantly larger within a class than between classes. As a result, the equations for the normal modes in the same class as the initially excited one can be approximated by driven harmonic oscillators, therefore, they get energy almost instantaneously. Because of the hierarchical organization of the mode coupling, the energy distribution among the modes will arrive at a stable profile, where most of the energy is localized on a few modes, leading to the formation of "natural package" and metastable states. The dynamics for modes in other symmetry classes follows a Mathieu type of equation, thus, interclass energy flow, when the initial excitation energy is small, starts typically when there is a mode that lies in the unstable region in the parameter space of Mathieu equation. Due to strong coupling of the modes inside the class, the whole class will get energy and be lifted up by the unstable mode. This characterizes the energy flow pathway of the system. These results bring fundamental understandings to the Fermi-Pasta-Ulam problem in two-dimensional systems with complex potentials, and reveal clearly the physical picture of dynamical interactions between the flexural modes, which will be crucial to the understanding of their abnormal contribution to heat conduction and nonlinear mechanical vibrations.
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Affiliation(s)
- Yisen Wang
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Zhigang Zhu
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Yong Zhang
- Department of Physics, Xiamen University, Xiamen 361005, China
| | - Liang Huang
- School of Physical Science and Technology, and Key Laboratory for Magnetism and Magnetic Materials of MOE, Lanzhou University, Lanzhou, Gansu 730000, China
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14
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Danieli C, Campbell DK, Flach S. Intermittent many-body dynamics at equilibrium. Phys Rev E 2017; 95:060202. [PMID: 28709247 DOI: 10.1103/physreve.95.060202] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2016] [Indexed: 06/07/2023]
Abstract
The equilibrium value of an observable defines a manifold in the phase space of an ergodic and equipartitioned many-body system. A typical trajectory pierces that manifold infinitely often as time goes to infinity. We use these piercings to measure both the relaxation time of the lowest frequency eigenmode of the Fermi-Pasta-Ulam chain, as well as the fluctuations of the subsequent dynamics in equilibrium. The dynamics in equilibrium is characterized by a power-law distribution of excursion times far off equilibrium, with diverging variance. Long excursions arise from sticky dynamics close to q-breathers localized in normal mode space. Measuring the exponent allows one to predict the transition into nonergodic dynamics. We generalize our method to Klein-Gordon lattices where the sticky dynamics is due to discrete breathers localized in real space.
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Affiliation(s)
- C Danieli
- New Zealand Institute for Advanced Study, Centre for Theoretical Chemistry & Physics, Massey University, Auckland 0745, New Zealand
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
| | - D K Campbell
- Department of Physics, Boston University, Boston, Massachusetts 02215, USA
| | - S Flach
- New Zealand Institute for Advanced Study, Centre for Theoretical Chemistry & Physics, Massey University, Auckland 0745, New Zealand
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
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15
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Ming Y, Ling DB, Li HM, Ding ZJ. Energy thresholds of discrete breathers in thermal equilibrium and relaxation processes. CHAOS (WOODBURY, N.Y.) 2017; 27:063106. [PMID: 28679219 DOI: 10.1063/1.4985016] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
So far, only the energy thresholds of single discrete breathers in nonlinear Hamiltonian systems have been analytically obtained. In this work, the energy thresholds of discrete breathers in thermal equilibrium and the energy thresholds of long-lived discrete breathers which can remain after a long time relaxation are analytically estimated for nonlinear chains. These energy thresholds are size dependent. The energy thresholds of discrete breathers in thermal equilibrium are the same as the previous analytical results for single discrete breathers. The energy thresholds of long-lived discrete breathers in relaxation processes are different from the previous results for single discrete breathers but agree well with the published numerical results known to us. Because real systems are either in thermal equilibrium or in relaxation processes, the obtained results could be important for experimental detection of discrete breathers.
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Affiliation(s)
- Yi Ming
- School of Physics and Material Science, Anhui University, Hefei, Anhui 230601, People's Republic of China
| | - Dong-Bo Ling
- School of Physics and Material Science, Anhui University, Hefei, Anhui 230601, People's Republic of China
| | - Hui-Min Li
- Supercomputing Center, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
| | - Ze-Jun Ding
- Department of Physics and Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, Anhui 230026, People's Republic of China
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16
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Aoki K. Stable and unstable periodic orbits in the one-dimensional lattice ϕ^{4} theory. Phys Rev E 2016; 94:042209. [PMID: 27841572 DOI: 10.1103/physreve.94.042209] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/10/2016] [Indexed: 11/07/2022]
Abstract
Periodic orbits for the classical ϕ^{4} theory on the one-dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, free and fixed boundary conditions. Through the process, we investigate which normal modes of the linear theory can or cannot be extended to the full nonlinear theory and why. We then analyze the stability of these orbits, clarifying the link between the stability, parametric resonance, and Lyapunov spectra for these orbits. The construction of the periodic orbits and the stability analysis is applicable to theories governed by Hamiltonians with quadratic intersite potentials and a general on-site potential. We also apply the analysis to theories with on-site potentials that have qualitatively different behavior from the ϕ^{4} theory, with some concrete examples.
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Affiliation(s)
- Kenichiro Aoki
- Research and Education Center for Natural Sciences and Hiyoshi Department of Physics, Keio University, Yokohama 223-8521, Japan
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17
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Green SR, Maillard A, Lehner L, Liebling SL. Islands of stability and recurrence times in AdS. Int J Clin Exp Med 2015. [DOI: 10.1103/physrevd.92.084001] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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18
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Buchel A, Green SR, Lehner L, Liebling SL. Conserved quantities and dual turbulent cascades in anti–de Sitter spacetime. Int J Clin Exp Med 2015. [DOI: 10.1103/physrevd.91.064026] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/07/2022]
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19
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Miloshevich G, Nguenang JP, Dauxois T, Khomeriki R, Ruffo S. Instabilities and relaxation to equilibrium in long-range oscillator chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:032927. [PMID: 25871192 DOI: 10.1103/physreve.91.032927] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/03/2014] [Indexed: 06/04/2023]
Abstract
We study instabilities and relaxation to equilibrium in a long-range extension of the Fermi-Pasta-Ulam-Tsingou (FPU) oscillator chain by exciting initially the lowest Fourier mode. Localization in mode space is stronger for the long-range FPU model. This allows us to uncover the sporadic nature of instabilities, i.e., by varying initially the excitation amplitude of the lowest mode, which is the control parameter, instabilities occur in narrow amplitude intervals. Only for sufficiently large values of the amplitude, the system enters a permanently unstable regime. These findings also clarify the long-standing problem of the relaxation to equilibrium in the short-range FPU model. Because of the weaker localization in mode space of this latter model, the transfer of energy is retarded and relaxation occurs on a much longer timescale.
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Affiliation(s)
- George Miloshevich
- Department of Physics, Faculty of Exact and Natural Sciences, Tbilisi State University, 0128 Tbilisi, Georgia
- Department of Physics, The University of Texas at Austin, Austin TX 78712, USA
| | - Jean-Pierre Nguenang
- Fundamental Physics Laboratory: Group of Nonlinear Physics and Complex Systems, Department of Physics, University of Douala, P.O. Box 24157, Douala, Cameroon
- Laboratoire de Physique de l'ENS Lyon, Université de Lyon, CNRS, 46, allée d'Italie, 69007 Lyon, France
| | - Thierry Dauxois
- Laboratoire de Physique de l'ENS Lyon, Université de Lyon, CNRS, 46, allée d'Italie, 69007 Lyon, France
| | - Ramaz Khomeriki
- Department of Physics, Faculty of Exact and Natural Sciences, Tbilisi State University, 0128 Tbilisi, Georgia
- Dipartimento di Fisica e Astronomia and CSDC, Università di Firenze, CNISM and INFN, via G. Sansone, 1, Sesto Fiorentino, Italy
| | - Stefano Ruffo
- Dipartimento di Fisica e Astronomia and CSDC, Università di Firenze, CNISM and INFN, via G. Sansone, 1, Sesto Fiorentino, Italy
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20
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Midtvedt D, Croy A, Isacsson A, Qi Z, Park HS. Fermi-Pasta-Ulam physics with nanomechanical graphene resonators: intrinsic relaxation and thermalization from flexural mode coupling. PHYSICAL REVIEW LETTERS 2014; 112:145503. [PMID: 24765986 DOI: 10.1103/physrevlett.112.145503] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2013] [Indexed: 06/03/2023]
Abstract
Thermalization in nonlinear systems is a central concept in statistical mechanics and has been extensively studied theoretically since the seminal work of Fermi, Pasta, and Ulam. Using molecular dynamics and continuum modeling of a ring-down setup, we show that thermalization due to nonlinear mode coupling intrinsically limits the quality factor of nanomechanical graphene drums and turns them into potential test beds for Fermi-Pasta-Ulam physics. We find the thermalization rate Γ to be independent of radius and scaling as Γ∼T*/εpre2, where T* and εpre are effective resonator temperature and prestrain.
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Affiliation(s)
- Daniel Midtvedt
- Department of Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
| | - Alexander Croy
- Department of Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
| | - Andreas Isacsson
- Department of Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden
| | - Zenan Qi
- Department of Mechanical Engineering, Boston University, Boston, Massachsetts 02215, USA
| | - Harold S Park
- Department of Mechanical Engineering, Boston University, Boston, Massachsetts 02215, USA
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21
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Cao LS, Qi DX, Peng RW, Wang M, Schmelcher P. Phononic frequency combs through nonlinear resonances. PHYSICAL REVIEW LETTERS 2014; 112:075505. [PMID: 24579614 DOI: 10.1103/physrevlett.112.075505] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/24/2013] [Indexed: 06/03/2023]
Abstract
We explore an analogue of optical frequency combs in driven nonlinear phononic systems, and present a mechanism for generating phononic frequency combs through nonlinear resonances. In the underlying process, a set of phonon modes is simultaneously excited by the external driving which yields frequency combs with an array of discrete and equidistant spectral lines of each nonlinearly excited phonon mode. Frequency combs through nonlinear resonance of different orders are investigated, and in particular the possibility of correlation tailoring in higher-order cases is revealed. We suggest that our results can be applied in various nonlinear acoustic processes, such as phonon harvesting, and can also be generalized to other nonlinear systems.
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Affiliation(s)
- L S Cao
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China and Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany and The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, D-22761 Hamburg, Germany
| | - D X Qi
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
| | - R W Peng
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
| | - Mu Wang
- National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China
| | - P Schmelcher
- Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany and The Hamburg Centre for Ultrafast Imaging, Luruper Chaussee 149, D-22761 Hamburg, Germany
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22
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Ponno A, Christodoulidi H, Skokos C, Flach S. The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior. CHAOS (WOODBURY, N.Y.) 2011; 21:043127. [PMID: 22225364 DOI: 10.1063/1.3658620] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
A numerical and analytical study of the relaxation to equilibrium of both the Fermi-Pasta-Ulam (FPU) α-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avalanche process. At the end of this first stage of the dynamics, the time-averaged modal energy spectrum of the Toda system stabilizes to its final profile, well described, at low energy, by the spectrum of a q-breather. The Toda equilibrium state is clearly shown to describe well the long-living quasi-state of the FPU system. On the long term, the modal energy spectrum of the FPU system slowly detaches from the Toda one by a diffusive-like rising of the tail modes, and eventually reaches the equilibrium flat shape. We find a simple law describing the growth of tail modes, which enables us to estimate the time-scale to equipartition of the FPU system, even when, at small energies, it becomes unobservable.
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Affiliation(s)
- A Ponno
- Università degli Studi di Padova, Dipartimento di Matematica Pura e Applicata, Via Trieste 63, 35121 Padova, Italy.
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23
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Manevitch LI, Smirnov VV. Limiting phase trajectories and the origin of energy localization in nonlinear oscillatory chains. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 82:036602. [PMID: 21230198 DOI: 10.1103/physreve.82.036602] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/24/2009] [Revised: 06/07/2010] [Indexed: 05/30/2023]
Abstract
We demonstrate that the modulation instability of the zone-boundary mode in a finite (periodic) Fermi-Pasta-Ulam chain is the necessary but not sufficient condition for the efficient energy transfer by localized excitations. This transfer results from the exclusion of complete energy exchange between spatially different parts of the chain, and the excitation level corresponding to that turns out to be twice more than threshold of zone-boundary mode's instability. To obtain this result one needs in far going extension of the beating concept to a wide class of finite oscillatory chains. In turn, such an extension leads to description of energy exchange and transition to energy localization and transfer in terms of effective particles and limiting phase trajectories. The effective particles appear naturally when the frequency spectrum crowding ensures the resonance interaction between zone-boundary and two nearby nonlinear normal modes, but there are no additional resonances. We show that the limiting phase trajectories corresponding to the most intensive energy exchange between effective particles can be considered as an alternative to nonlinear normal modes, which describe the stationary process.
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Affiliation(s)
- L I Manevitch
- Institute of Chemical Physics, RAS, 4 Kosygin Str., Moscow 119991, Russia
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24
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Djoufack ZI, Kenfack-Jiotsa A, Nguenang JP, Domngang S. Quantum signatures of breathers in a finite Heisenberg spin chain. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2010; 22:205502. [PMID: 21393707 DOI: 10.1088/0953-8984/22/20/205502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
A map of a quantum Heisenberg spin chain into an extended Bose-Hubbard-like Hamiltonian is set up. Within this framework, the spectrum of the corresponding Bose-Hubbard chain, on a periodic one-dimensional lattice containing two, four, and six bosons shows interesting detailed band structures. These fine structures are studied using numerical diagonalization, and nondegenerate and degenerate perturbation theory. We also focus our attention on the effect of the anisotropy and Heisenberg exchange energy on the detailed band structures. The signature of the quantum breather is also set up by the square of the amplitudes of the corresponding eigenvectors in real space.
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Affiliation(s)
- Z I Djoufack
- Nonlinear Physics and Complex Systems Group, Département de Physique, Ecole Normale Supérieure, Université de Yaoundé I, PO Box 47, Yaoundé, Cameroon
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25
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Christodoulidi H, Efthymiopoulos C, Bountis T. Energy localization on q-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2010; 81:016210. [PMID: 20365449 DOI: 10.1103/physreve.81.016210] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/24/2009] [Revised: 09/24/2009] [Indexed: 05/29/2023]
Abstract
We focus on two approaches that have been proposed in recent years for the explanation of the so-called Fermi-Pasta-Ulam (FPU) paradox, i.e., the persistence of energy localization in the "low-q " Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of "natural packets" exhibiting exponential stability, while in the second, emphasis is placed on the existence of "q breathers," i.e., periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of " q-tori" representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.
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Affiliation(s)
- H Christodoulidi
- Department of Mathematics, University of Patras, Patras, Greece.
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26
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Ivanchenko MV. q Breathers in finite lattices: nonlinearity and weak disorder. PHYSICAL REVIEW LETTERS 2009; 102:175507. [PMID: 19518797 DOI: 10.1103/physrevlett.102.175507] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/03/2008] [Indexed: 05/27/2023]
Abstract
Nonlinearity and disorder are the recognized ingredients of the lattice vibrational dynamics, the factors that could be diminished, but never excluded. We generalize the concept of q breathers-periodic orbits in nonlinear lattices, exponentially localized in the linear mode space-to the case of weak disorder, taking the Fermi-Pasta-Ulan chain as an example. We show that these nonlinear vibrational modes remain exponentially localized near the central mode and stable, provided the disorder is sufficiently small. The instability threshold depends sensitively on a particular realization of disorder and can be modified by specifically designed impurities. Based on this sensitivity, an approach to controlling the energy flow between the modes is proposed. The relevance to other model lattices and experimental miniature arrays is discussed.
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Affiliation(s)
- M V Ivanchenko
- Department of Applied Mathematics, University of Leeds, LS2 9JT, Leeds, United Kingdom
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27
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Shrestha U, Kostrun M, Javanainen J. Pulsating instability of a Bose-Einstein condensate in an optical lattice. PHYSICAL REVIEW LETTERS 2008; 101:070406. [PMID: 18764516 DOI: 10.1103/physrevlett.101.070406] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/05/2008] [Indexed: 05/26/2023]
Abstract
We find numerically that in the limit of weak atom-atom interactions a Bose-Einstein condensate in an optical lattice may develop a pulsating dynamical instability in which the atoms nearly periodically form a peak in the occupation numbers of the lattice sites, and then return to the unstable initial state. Multiple peaks behaving similarly are also found. Simple arguments show that the pulsating instability is a remnant of integrability, and give a handle on the relevant physical scales.
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Affiliation(s)
- Uttam Shrestha
- Department of Physics, University of Connecticut, Storrs, Connecticut 06269-3046, USA
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28
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Jia S, Wan W, Fleischer JW. Forward four-wave mixing with defocusing nonlinearity. OPTICS LETTERS 2007; 32:1668-70. [PMID: 17572741 DOI: 10.1364/ol.32.001668] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/15/2023]
Abstract
We experimentally demonstrate degenerate, forward four-wave mixing effects in a self-defocusing photorefractive medium, in both one and two transverse dimensions. We observe the nonlinear evolution of new modes as a function of propagation distance, in both the near-field and far-field (Fourier space) regions.
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Affiliation(s)
- Shu Jia
- Department of Electrical Engineering, Princeton University, NJ 08544, USA
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29
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Penati T, Flach S. Tail resonances of Fermi-Pasta-Ulam q-breathers and their impact on the pathway to equipartition. CHAOS (WOODBURY, N.Y.) 2007; 17:023102. [PMID: 17614656 DOI: 10.1063/1.2645141] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
Upon initial excitation of a few normal modes the energy distribution among all modes of a nonlinear atomic chain (the Fermi-Pasta-Ulam model) exhibits exponential localization on large time scales. At the same time, resonant anomalies (peaks) are observed in its weakly excited tail for long times preceding equipartition. We observe a similar resonant tail structure also for exact time-periodic Lyapunov orbits, coined q-breathers due to their exponential localization in modal space. We give a simple explanation for this structure in terms of superharmonic resonances. The resonance analysis agrees very well with numerical results and has predictive power. We extend a previously developed perturbation method, based essentially on a Poincare-Lindstedt scheme, in order to account for these resonances, and in order to treat more general model cases, including truncated Toda potentials. Our results give a qualitative and semiquantitative account for the superharmonic resonances of q-breathers and natural packets.
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Affiliation(s)
- Tiziano Penati
- Dipartimento di Matematica F. Enriques, Via Saldini 50, 20133 Milano, Italy.
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30
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Sato M, Sievers AJ. Driven localized excitations in the acoustic spectrum of small nonlinear macroscopic and microscopic lattices. PHYSICAL REVIEW LETTERS 2007; 98:214101. [PMID: 17677775 DOI: 10.1103/physrevlett.98.214101] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/01/2006] [Indexed: 05/16/2023]
Abstract
Both bright and dark traveling, locked, intrinsic localized modes (ILMs) have been generated with a spatially uniform driver at a frequency in the acoustic spectrum of a nonlinear micromechanical cantilever array. Complementary numerical simulations show that a minimum density of modes, hence array size, is required for the formation of such locked smoothly running excitations. Additional simulations on a small 1D antiferromagnetic spin system are used to illustrate that such uniformly driven running ILMs should be a generic feature of a nanoscale atomic lattice.
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Affiliation(s)
- M Sato
- Graduate School of Natural Science and Technology, Kanazawa University, Kanazawa, Ishikawa 920-1192, Japan
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31
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Gorbach AV, Skryabin DV. Spectral-discrete solitons and localization in frequency space. OPTICS LETTERS 2006; 31:3309-11. [PMID: 17072406 DOI: 10.1364/ol.31.003309] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
We report families of discrete optical solitons in frequency space, or spectral-discrete solitons existing in a dispersive Raman medium, where individual sidebands are coupled by coherence. The associated time-domain patterns correspond either to trains of ultrashort pulses or to weakly modulated waves. We describe the physics behind the spectral localization and study soliton bifurcations, stability, and dynamics.
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Affiliation(s)
- A V Gorbach
- Centre for Photonics and Photonic Materials, Department of Physics, University of Bath, UK.
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32
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Ivanchenko MV, Kanakov OI, Mishagin KG, Flach S. q-Breathers in finite two- and three-dimensional nonlinear acoustic lattices. PHYSICAL REVIEW LETTERS 2006; 97:025505. [PMID: 16907458 DOI: 10.1103/physrevlett.97.025505] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/28/2006] [Indexed: 05/11/2023]
Abstract
In their celebrated experiment, Fermi, Pasta, and Ulam (FPU) [Los Alamos Report No. LA-1940, 1955] observed that in simple one-dimensional nonlinear atomic chains the energy must not always be equally shared among the modes. Recently, it was shown that exact and stable time-periodic orbits, coined q-breathers (QBs), localize the mode energy in normal mode space in an exponential way, and account for many aspects of the FPU problem. Here we take the problem into more physically important cases of two- and three-dimensional acoustic lattices to find existence and principally different features of QBs. By use of perturbation theory and numerical calculations we obtain that the localization and stability of QBs are enhanced with increasing system size in higher lattice dimensions opposite to their one-dimensional analogues.
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Affiliation(s)
- M V Ivanchenko
- Department of Radiophysics, Nizhny Novgorod University, Gagarin Avenue 23, 603950 Nizhny Novgorod, Russia
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33
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Antonopoulos C, Bountis T. Stability of simple periodic orbits and chaos in a Fermi-Pasta-Ulam lattice. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:056206. [PMID: 16803025 DOI: 10.1103/physreve.73.056206] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/18/2006] [Indexed: 05/10/2023]
Abstract
We investigate the connection between local and global dynamics in the Fermi-Pasta-Ulam (FPU) beta model from the point of view of stability of its simplest periodic orbits (SPO's). In particular, we show that there is a relatively high-q mode [q = 2(N + 1)/3] of the linear lattice, having one particle fixed every two oppositely moving ones (called SPO2 here), which can be exactly continued to the nonlinear case for N = 5 + 3m, m = 0,1,2,, and whose first destabilization E(2u), as the energy (or beta) increases for any fixed N, practically coincides with the onset of a "weak" form of chaos preceding the breakdown of FPU recurrences, as predicted recently in a similar study of the continuation of a very low (q = 3) mode of the corresponding linear chain. This energy threshold per particle behaves like E(2u)/N alpha N(-2). We also follow exactly the properties of another SPO [with q = (N + 1)/2] in which fixed and moving particles are interchanged (called SPO1 here) and which destabilizes at higher energies than SPO2, since E(1u)/N alpha N(-1). We find that, immediately after their first destabilization, these SPO's have different (positive) Lyapunov spectra in their vicinity. However, as the energy increases further (at fixed N), these spectra converge to the same exponentially decreasing function, thus providing strong evidence that the chaotic regions around SPO1 and SPO2 have "merged" and large-scale chaos has spread throughout the lattice. Since these results hold for N arbitrarily large, they suggest a direct approach by which one can use local stability analysis of SPO's to estimate the energy threshold at which a transition to ergodicity occurs and thermodynamic properties such as Kolmogorov-Sinai entropies per particle can be computed for similar one-dimensional lattices.
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Affiliation(s)
- Chris Antonopoulos
- Department of Mathematics and Center for Research and Applications of Nonlinear Systems (CRANS), University of Patras, Greece
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34
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Ponno A, Ruggiero J, Drigo E, De Luca J. Energy localization in the phi4 oscillator chain. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:056609. [PMID: 16803059 DOI: 10.1103/physreve.73.056609] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/08/2006] [Indexed: 05/10/2023]
Abstract
We study energy localization in a finite one-dimensional phi(4) oscillator chain with initial energy in a single oscillator of the chain. We numerically calculate the effective number of degrees of freedom sharing the energy on the lattice as a function of time. We find that for energies smaller than a critical value, energy equipartition among the oscillators is reached in a relatively short time. On the other hand, above the critical energy, a decreasing number of particles sharing the energy is observed. We give an estimate of the effective number of degrees of freedom as a function of the energy. Our results suggest that localization is due to the appearance, above threshold, of a breather-like structure. Analytic arguments are given, based on the averaging theory and the analysis of a discrete nonlinear Schrödinger equation approximating the dynamics, to support and explain the numerical results.
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Affiliation(s)
- A Ponno
- Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, Italy.
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Maniadis P, Bountis T. Quasiperiodic and chaotic discrete breathers in a parametrically driven system without linear dispersion. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:046211. [PMID: 16711921 DOI: 10.1103/physreve.73.046211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/07/2005] [Revised: 02/24/2006] [Indexed: 05/09/2023]
Abstract
We study a one-dimensional lattice of anharmonic oscillators with only quartic nearest-neighbor interactions, in which discrete breathers (DB's) can be explicitly constructed by an exact separation of their time and space dependence. Introducing parametric periodic driving, we first show how a variety of such DB's can be obtained by selecting spatial profiles from the homoclinic orbits of an invertible map and combining them with initial conditions chosen from the Poincaré surface of section of a simple Duffing's equation. Placing then our initial conditions at the center of the islands of a major resonance, we demonstrate how the corresponding DB can be stabilized by varying the amplitude of the driving. We thus discover around elliptic points a large region of quasiperiodic breathers, which are stable for very long times. Starting with initial conditions close to the elliptic point at the origin, we find that as we approach the main chaotic layer, a quasiperiodic breather either destabilizes by delocalization or turns into a chaotic breather, with an evidently broadbanded Fourier spectrum before it collapses. For some breather profiles stable quasiperiodic breathers exist all the way to the separatrix of the Duffing equation, indicating the presence of large regions of tori around the DB solution in the multidimensional phase space. We argue that these strong localization phenomena are due to the absence of phonon resonances, as there are no linear dispersion terms in our lattices. We also show, however, that these phenomena persist in more realistic physical models, in which weak linear dispersion is included in the equations of motion, with a sufficiently small coefficient.
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Affiliation(s)
- P Maniadis
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, D-01187 Dresden, Germany
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Flach S, Ivanchenko MV, Kanakov OI. q-breathers in Fermi-Pasta-Ulam chains: existence, localization, and stability. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2006; 73:036618. [PMID: 16605688 DOI: 10.1103/physreve.73.036618] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/30/2005] [Indexed: 05/08/2023]
Abstract
The Fermi-Pasta-Ulam (FPU) problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model. In the harmonic limit, each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q. We continue normal modes from the harmonic limit into the FPU parameter regime and obtain persistence of these periodic orbits, termed here q-breathers (QB). They are characterized by time periodicity, exponential localization in the q-space of normal modes, and linear stability up to a size-dependent threshold amplitude. Trajectories computed in the original FPU setting are perturbations around these exact QB solutions. The QB concept is applicable to other nonlinear lattices as well.
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Affiliation(s)
- S Flach
- Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany
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