1
|
Liu Y, He D. Chaotic route to classical thermalization: A real-space analysis. Phys Rev E 2024; 109:064115. [PMID: 39020962 DOI: 10.1103/physreve.109.064115] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/07/2024] [Accepted: 05/23/2024] [Indexed: 07/20/2024]
Abstract
Most of the previous studies on classical thermalization focus on the wave-vector space, encountering limitations when extended beyond quasi-integrable regions. In this study, we propose a scheme to study the thermalization of the classical Hamiltonian chain of interacting oscillators in real space by developing a thermalization indicator proposed by Parisi [Europhys. Lett. 40, 357 (1997)0295-507510.1209/epl/i1997-00471-9], which approaches zero in the thermal state. Upon reaching the steady state characterized by the generalized Gibbs ensemble for a harmonic chain, a quench protocol is implemented to change the Hamiltonian to a nonintegrable form instantaneously, thereby preparing nonequilibrium initial states. This approach enables investigations of thermalization in real space, particularly valuable for exploring regions beyond quasi-integrability. For the FPUT-β lattice, we observe that the thermalization time as a function of the nonintegrable strength follows a -2 scaling law in the quasi-integrable region and -1/4 in the strongly integrable region. Moreover, numerical results reveal the thermalization time is proportional to the Lyapunov time, which bridges microscopic chaotic dynamics and the macroscopic thermalization process.
Collapse
|
2
|
Wang Z, Fu W, Zhang Y, Zhao H. Thermalization of Two- and Three-Dimensional Classical Lattices. PHYSICAL REVIEW LETTERS 2024; 132:217102. [PMID: 38856278 DOI: 10.1103/physrevlett.132.217102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/03/2022] [Revised: 02/11/2024] [Accepted: 05/02/2024] [Indexed: 06/11/2024]
Abstract
Understanding how systems achieve thermalization is a fundamental task in statistical physics. This Letter presents both analytical and numerical evidence showing that thermalization can be universally achieved in sufficiently large two- and three-dimensional lattices via weak nonlinear interactions. Thermalization time follows a universal scaling law unaffected by lattice structures, types of interaction potentials, or whether the lattice is ordered or not. Moreover, this study highlights the critical impact of dimensionality and degeneracy on thermalization dynamics.
Collapse
Affiliation(s)
- Zhen Wang
- Department of Physics, Xiamen University, Xiamen 361005, Fujian, China
- CAS Key Laboratory of Theoretical Physics and Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
| | - Weicheng Fu
- Department of Physics, Tianshui Normal University, Tianshui 741001, Gansu, China
- Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Yong Zhang
- Department of Physics, Xiamen University, Xiamen 361005, Fujian, China
- Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| | - Hong Zhao
- Department of Physics, Xiamen University, Xiamen 361005, Fujian, China
- Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
| |
Collapse
|
3
|
Mithun T, Maluckov A, Mančić A, Khare A, Kevrekidis PG. How close are integrable and nonintegrable models: A parametric case study based on the Salerno model. Phys Rev E 2023; 107:024202. [PMID: 36932573 DOI: 10.1103/physreve.107.024202] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/10/2022] [Accepted: 01/09/2023] [Indexed: 06/18/2023]
Abstract
In the present work we revisit the Salerno model as a prototypical system that interpolates between a well-known integrable system (the Ablowitz-Ladik lattice) and an experimentally tractable, nonintegrable one (the discrete nonlinear Schrödinger model). The question we ask is, for "generic" initial data, how close are the integrable to the nonintegrable models? Our more precise formulation of this question is, How well is the constancy of formerly conserved quantities preserved in the nonintegrable case? Upon examining this, we find that even slight deviations from integrability can be sensitively felt by measuring these formerly conserved quantities in the case of the Salerno model. However, given that the knowledge of these quantities requires a deep physical and mathematical analysis of the system, we seek a more "generic" diagnostic towards a manifestation of integrability breaking. We argue, based on our Salerno model computations, that the full spectrum of Lyapunov exponents could be a sensitive diagnostic to that effect.
Collapse
Affiliation(s)
- Thudiyangal Mithun
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - Aleksandra Maluckov
- COHERENCE, Vinča Institute of Nuclear Sciences, National Institute of the Republic of Serbia, University of Belgrade, P.O.B. 522, 11001 Belgrade, Republic of Serbia
| | - Ana Mančić
- COHERENCE, Department of Physics, Faculty of Sciences and Mathematics, University of Niš, P.O.B. 224, 18000 Niš, Serbia
| | - Avinash Khare
- Department of Physics, Savitribai Phule Pune University, Pune 411007, India
| | - Panayotis G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| |
Collapse
|
4
|
McRoberts AJ, Bilitewski T, Haque M, Moessner R. Long-lived solitons and their signatures in the classical Heisenberg chain. Phys Rev E 2022; 106:L062202. [PMID: 36671135 DOI: 10.1103/physreve.106.l062202] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2022] [Accepted: 11/28/2022] [Indexed: 12/23/2022]
Abstract
Motivated by the Kardar-Parisi-Zhang (KPZ) scaling recently observed in the classical ferromagnetic Heisenberg chain, we investigate the role of solitonic excitations in this model. We find that the Heisenberg chain, although well known to be nonintegrable, supports a two-parameter family of long-lived solitons. We connect these to the exact soliton solutions of the integrable Ishimori chain with ln(1+S_{i}·S_{j}) interactions. We explicitly construct infinitely long-lived stationary solitons, and provide an adiabatic construction procedure for moving soliton solutions, which shows that Ishimori solitons have a long-lived Heisenberg counterpart when they are not too narrow and not too fast moving. Finally, we demonstrate their presence in thermal states of the Heisenberg chain, even when the typical soliton width is larger than the spin correlation length, and argue that these excitations likely underlie the KPZ scaling.
Collapse
Affiliation(s)
- Adam J McRoberts
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| | - Thomas Bilitewski
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany.,Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA
| | - Masudul Haque
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany.,Department of Theoretical Physics, Maynooth University, County Kildare, Ireland.,Institut für Theoretische Physik, Technische Universität Dresden, 01062 Dresden, Germany
| | - Roderich Moessner
- Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Strasse 38, 01187 Dresden, Germany
| |
Collapse
|
5
|
Malishava M, Flach S. Thermalization dynamics of macroscopic weakly nonintegrable maps. CHAOS (WOODBURY, N.Y.) 2022; 32:063113. [PMID: 35778154 DOI: 10.1063/5.0092032] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/20/2022] [Accepted: 05/16/2022] [Indexed: 06/15/2023]
Abstract
We study thermalization of weakly nonintegrable nonlinear unitary lattice dynamics. We identify two distinct thermalization regimes close to the integrable limits of either linear dynamics or disconnected lattice dynamics. For weak nonlinearity, the almost conserved actions correspond to extended observables which are coupled into a long-range network. For weakly connected lattices, the corresponding local observables are coupled into a short-range network. We compute the evolution of the variance σ ( T ) of finite time average distributions for extended and local observables. We extract the ergodization time scale T which marks the onset of thermalization, and determine the type of network through the subsequent decay of σ ( T ). We use the complementary analysis of Lyapunov spectra [M. Malishava and S. Flach, Phys. Rev. Lett. 128, 134102 (2022)] and compare the Lyapunov time T with T. We characterize the spatial properties of the tangent vector and arrive at a complete classification picture of weakly nonintegrable macroscopic thermalization dynamics.
Collapse
Affiliation(s)
- Merab Malishava
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, South Korea
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, South Korea
| |
Collapse
|
6
|
Malishava M, Flach S. Lyapunov Spectrum Scaling for Classical Many-Body Dynamics Close to Integrability. PHYSICAL REVIEW LETTERS 2022; 128:134102. [PMID: 35426693 DOI: 10.1103/physrevlett.128.134102] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/16/2021] [Revised: 12/02/2021] [Accepted: 03/08/2022] [Indexed: 05/29/2023]
Abstract
We propose a novel framework to characterize the thermalization of many-body dynamical systems close to integrable limits using the scaling properties of the full Lyapunov spectrum. We use a classical unitary map model to investigate macroscopic weakly nonintegrable dynamics beyond the limits set by the KAM regime. We perform our analysis in two fundamentally distinct long-range and short-range integrable limits which stem from the type of nonintegrable perturbations. Long-range limits result in a single parameter scaling of the Lyapunov spectrum, with the inverse largest Lyapunov exponent being the only diverging control parameter and the rescaled spectrum approaching an analytical function. Short-range limits result in a dramatic slowing down of thermalization which manifests through the rescaled Lyapunov spectrum approaching a non-analytic function. An additional diverging length scale controls the exponential suppression of all Lyapunov exponents relative to the largest one.
Collapse
Affiliation(s)
- Merab Malishava
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea and Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Korea
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon 34126, Korea and Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Korea
| |
Collapse
|
7
|
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.
Collapse
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
| |
Collapse
|
8
|
Hahn D, Urbina JD, Richter K, Dubertrand R, Sondhi SL. Ergodic and nonergodic many-body dynamics in strongly nonlinear lattices. Phys Rev E 2021; 103:052213. [PMID: 34134210 DOI: 10.1103/physreve.103.052213] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2020] [Accepted: 04/22/2021] [Indexed: 11/07/2022]
Abstract
The study of nonlinear oscillator chains in classical many-body dynamics has a storied history going back to the seminal work of Fermi et al. [Los Alamos Scientific Laboratory Report No. LA-1940, 1955 (unpublished)]. We introduce a family of such systems which consist of chains of N harmonically coupled particles with the nonlinearity introduced by confining the motion of each individual particle to a box or stadium with hard walls. The stadia are arranged on a one-dimensional lattice but they individually do not have to be one dimensional, thus permitting the introduction of chaos already at the lattice scale. For the most part we study the case where the motion is entirely one dimensional. We find that the system exhibits a mixed phase space for any finite value of N. Computations of Lyapunov spectra at randomly picked phase space locations and a direct comparison between Hamiltonian evolution and phase space averages indicate that the regular regions of phase space are not significant at large system sizes. While the continuum limit of our model is itself a singular limit of the integrable sinh Gordon theory, we do not see any evidence for the kind of nonergodicity famously seen in the work of Fermi et al. Finally, we examine the chain with particles confined to two-dimensional stadia where the individual stadium is already chaotic and find a much more chaotic phase space at small system sizes.
Collapse
Affiliation(s)
- Dominik Hahn
- Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany.,Max Planck Institute for the Physics of Complex Systems, Noethnitzer Straße 38, 01187 Dresden, Germany
| | - Juan-Diego Urbina
- Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany
| | - Klaus Richter
- Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany
| | - Rémy Dubertrand
- Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany.,Department of Mathematics, Physics and Electrical Engineering, Northumbria University, NE1 8ST Newcastle upon Tyne, United Kingdom
| | - S L Sondhi
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| |
Collapse
|
9
|
Mithun T, Maluckov A, Manda BM, Skokos C, Bishop A, Saxena A, Khare A, Kevrekidis PG. Thermalization in the one-dimensional Salerno model lattice. Phys Rev E 2021; 103:032211. [PMID: 33862787 DOI: 10.1103/physreve.103.032211] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/01/2020] [Accepted: 03/04/2021] [Indexed: 11/07/2022]
Abstract
The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.
Collapse
Affiliation(s)
- Thudiyangal Mithun
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| | - Aleksandra Maluckov
- Vinca Institute of Nuclear Sciences, University of Belgrade, National Institute of the Republic of Serbia, P.O.B. 522, 11001 Belgrade, Serbia.,Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, S. Korea
| | - Bertin Many Manda
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - Charalampos Skokos
- Nonlinear Dynamics and Chaos Group, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - Alan Bishop
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Avadh Saxena
- Center for Nonlinear Studies and Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
| | - Avinash Khare
- Department of Physics, Savitribai Phule Pune University, Pune 411007, India
| | - Panayotis G Kevrekidis
- Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA
| |
Collapse
|
10
|
Liu Y, He D. Analytical approach to Lyapunov time: Universal scaling and thermalization. Phys Rev E 2021; 103:L040203. [PMID: 34005992 DOI: 10.1103/physreve.103.l040203] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/10/2021] [Accepted: 04/01/2021] [Indexed: 06/12/2023]
Abstract
Based on the geometrization of dynamics and self-consistent phonon theory, we develop an analytical approach to derive the Lyapunov time, the reciprocal of the largest Lyapunov exponent, for general nonlinear lattices of coupled oscillators. The Fermi-Pasta-Ulam-Tsingou-like lattices are exemplified by using the method, which agree well with molecular dynamical simulations for the cases of quartic and sextic interactions. A universal scaling behavior of the Lyapunov time with the nonintegrability strength is observed for the quasi-integrable regime. Interestingly, the scaling exponent of the Lyapunov time is the same as the thermalization time, which indicates a proportional relationship between the two timescales. This relation illustrates how the thermalization process is related to the intrinsic chaotic property.
Collapse
Affiliation(s)
- Yue Liu
- Department of Physics and Jiujiang Research Institute, Xiamen University, Xiamen 361005, Fujian, China
| | - Dahai He
- Department of Physics and Jiujiang Research Institute, Xiamen University, Xiamen 361005, Fujian, China
| |
Collapse
|
11
|
Wang Z, Fu W, Zhang Y, Zhao H. Wave-Turbulence Origin of the Instability of Anderson Localization against Many-Body Interactions. PHYSICAL REVIEW LETTERS 2020; 124:186401. [PMID: 32441955 DOI: 10.1103/physrevlett.124.186401] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2019] [Revised: 02/08/2020] [Accepted: 04/03/2020] [Indexed: 06/11/2023]
Abstract
Whether Anderson localization is robust against many-body interactions and, closely related, whether a disordered many-body system can be thermalized are long outstanding issues. In this Letter, we address these issues with the wave-turbulence theory. We show that, in general, the thermalization time in one-dimensional disordered lattice systems is inversely proportional to the squared interaction strength in the thermodynamic limit. It leads to the conclusion that such systems can always be thermalized by arbitrarily weak many-body interactions and thus the localized states are unstable.
Collapse
Affiliation(s)
- Zhen Wang
- Department of Physics, Xiamen University, Xiamen 361005, Fujian, China
| | - Weicheng Fu
- Department of Physics, Xiamen University, Xiamen 361005, Fujian, China
| | - Yong Zhang
- Department of Physics, Xiamen University, Xiamen 361005, Fujian, China
| | - Hong Zhao
- Department of Physics, Xiamen University, Xiamen 361005, Fujian, China
| |
Collapse
|