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De Roeck W, Huveneers F, Prośniak OA. Slow dissipation and spreading in disordered classical systems: A direct comparison between numerics and mathematical bounds. Phys Rev E 2024; 109:044207. [PMID: 38755865 DOI: 10.1103/physreve.109.044207] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2023] [Accepted: 01/29/2024] [Indexed: 05/18/2024]
Abstract
We study the breakdown of Anderson localization in the one-dimensional nonlinear Klein-Gordon chain, a prototypical example of a disordered classical many-body system. A series of numerical works indicate that an initially localized wave packet spreads polynomially in time, while analytical studies rather suggest a much slower spreading. Here, we focus on the decorrelation time in equilibrium. On the one hand, we provide a mathematical theorem establishing that this time is larger than any inverse power law in the effective anharmonicity parameter λ, and on the other hand our numerics show that it follows a power law for a broad range of values of λ. This numerical behavior is fully consistent with the power law observed numerically in spreading experiments, and we conclude that the state-of-the-art numerics may well be unable to capture the long-time behavior of such classical disordered systems.
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Affiliation(s)
| | - Francois Huveneers
- Department of Mathematics, King's College London, Strand, London WC2R 2LS, United Kingdom
| | - Oskar A Prośniak
- Department of Physics and Materials Science, University of Luxembourg, L-1511 Luxembourg, Grand Duchy of Luxembourg
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Roy D, Huse DA, Kulkarni M. Out-of-time-ordered correlator in the one-dimensional Kuramoto-Sivashinsky and Kardar-Parisi-Zhang equations. Phys Rev E 2023; 108:054112. [PMID: 38115452 DOI: 10.1103/physreve.108.054112] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/16/2023] [Accepted: 10/11/2023] [Indexed: 12/21/2023]
Abstract
The out-of-time-ordered correlator (OTOC) has emerged as an interesting object in both classical and quantum systems for probing the spatial spread and temporal growth of initially local perturbations in spatially extended chaotic systems. Here, we study the (classical) OTOC and its "light cone" in the nonlinear Kuramoto-Sivashinsky (KS) equation, using extensive numerical simulations. We also show that the linearized KS equation exhibits a qualitatively similar OTOC and light cone, which can be understood via a saddle-point analysis of the linearly unstable modes. Given the deep connection between the KS (deterministic) and the Kardar-Parisi-Zhang (KPZ, which is stochastic) equations, we also explore the OTOC in the KPZ equation. While our numerical results in the KS case are expected to hold in the continuum limit, for the KPZ case it is valid in a discretized version of the KPZ equation. More broadly, our work unravels the intrinsic interplay between noise/instability, nonlinearity, and dissipation in partial differential equations (deterministic or stochastic) through the lens of OTOC.
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Affiliation(s)
- Dipankar Roy
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
| | - David A Huse
- Physics Department, Princeton University, Princeton, New Jersey 08544, USA
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore 560089, India
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S BK, Huse DA, Kulkarni M. Spatiotemporal spread of perturbations in power-law models at low temperatures: Exact results for classical out-of-time-order correlators. Phys Rev E 2021; 104:044117. [PMID: 34781511 DOI: 10.1103/physreve.104.044117] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/03/2020] [Accepted: 09/23/2021] [Indexed: 11/07/2022]
Abstract
We present exact results for the classical version of the out-of-time-order commutator (OTOC) for a family of power-law models consisting of N particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form ∝∑_{i,j=1(i≠j)}^{N}|x_{i}-x_{j}|^{-k}∀k>1 where x_{i} is the position of the ith particle. We present numerical results for the OTOC for finite N at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many-body ground state. In the large-N limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the "light cone." We find remarkably distinct features in OTOC below and above k=3 in terms of going from non-Airy behavior (1<k<3) to an Airy universality class (k>3). We present certain additional rich features for the case k=2 that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatiotemporal spread of perturbations in long-range interacting systems.
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Affiliation(s)
- Bhanu Kiran S
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - David A Huse
- Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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Chatterjee AK, Kulkarni M, Kundu A. Dynamical regimes of finite-temperature discrete nonlinear Schrödinger chain. Phys Rev E 2021; 104:044136. [PMID: 34781584 DOI: 10.1103/physreve.104.044136] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/15/2021] [Accepted: 10/12/2021] [Indexed: 11/07/2022]
Abstract
We show that the one-dimensional discrete nonlinear Schrödinger chain (DNLS) at finite temperature has three different dynamical regimes (ultralow-, low-, and high-temperature regimes). This has been established via (i) one-point macroscopic thermodynamic observables (temperature T, energy density ε, and the relationship between them), (ii) emergence and disappearance of an additional almost conserved quantity (total phase difference), and (iii) classical out-of-time-ordered correlators and related quantities (butterfly speed and Lyapunov exponents). The crossover temperatures T_{l-ul} (between low- and ultra-low-temperature regimes) and T_{h-l} (between the high- and low-temperature regimes) extracted from these three different approaches are consistent with each other. The analysis presented here is an important step forward toward the understanding of DNLS which is ubiquitous in many fields and has a nonseparable Hamiltonian form. Our work also shows that the different methods used here can serve as important tools to identify dynamical regimes in other interacting many-body systems.
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Affiliation(s)
- Amit Kumar Chatterjee
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Manas Kulkarni
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
| | - Anupam Kundu
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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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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