1
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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.
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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
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2
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Gotti G, Iubini S, Politi P. Condensation induced by coupled transport processes. Phys Rev E 2022; 106:054158. [PMID: 36559399 DOI: 10.1103/physreve.106.054158] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/08/2022] [Accepted: 11/14/2022] [Indexed: 06/17/2023]
Abstract
Several lattice models display a condensation transition in real space when the density of a suitable order parameter exceeds a critical value. We consider one of such models with two conservation laws, in a onedimensional open setup where the system is attached to two external reservoirs. Both reservoirs impose subcritical boundary conditions at the chain ends. When such boundary conditions are equal, the system is in equilibrium below the condensation threshold and no condensate can appear. Instead, when the system is kept out of equilibrium, localization may arise in an internal portion of the lattice. We discuss the origin of this phenomenon, the relevance of the number of conservation laws, and the effect of the pinning of the condensate on the dynamics of the out-of-equilibrium state.
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Affiliation(s)
- Gabriele Gotti
- Dipartimento di Fisica e Astronomia, Università di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
- Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
| | - Stefano Iubini
- Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
| | - Paolo Politi
- Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, via Madonna del Piano 10, I-50019 Sesto Fiorentino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
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3
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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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4
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Bagchi D. Heat transport in long-ranged Fermi-Pasta-Ulam-Tsingou-type models. Phys Rev E 2021; 104:054108. [PMID: 34942781 DOI: 10.1103/physreve.104.054108] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/07/2021] [Accepted: 10/14/2021] [Indexed: 11/07/2022]
Abstract
We perform a detailed study of heat transport in one-dimensional long-ranged anharmonic oscillator systems, such as the long-ranged Fermi-Pasta-Ulam-Tsingou model. For these systems, the long-ranged anharmonic potential decays with distance as a power law, controlled by an exponent δ≥0. For such a nonintegrable model, one of the recent results that has captured quite some attention is the puzzling ballisticlike transport observed for δ=2, reminiscent of integrable systems. Here, we first employ the reverse nonequilibrium molecular dynamics simulations to look closely at the δ=2 transport in three long-ranged models and point out a few problematic issues with this simulation method. Next, we examine the process of energy relaxation, and find that relaxation can be appreciably slow for δ=2 in some situations. We invoke the concept of nonlinear localized modes of excitation, also known as discrete breathers, and demonstrate that the slow relaxation and the ballisticlike transport properties can be consistently explained in terms of a novel depinning of the discrete breathers that makes them highly mobile at δ=2. Finally, in the presence of quartic pinning potentials we find that the long-ranged model exhibits Fourier (diffusive) transport at δ=2, as one would expect from short-ranged interacting systems with broken momentum conservation. Such a diffusive regime is not observed for harmonic pinning.
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Affiliation(s)
- Debarshee Bagchi
- International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bengaluru 560089, India
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5
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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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6
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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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7
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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.
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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
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8
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Gradenigo G, Iubini S, Livi R, Majumdar SN. Condensation transition and ensemble inequivalence in the discrete nonlinear Schrödinger equation. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2021; 44:29. [PMID: 33710395 DOI: 10.1140/epje/s10189-021-00046-5] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/16/2020] [Accepted: 02/25/2021] [Indexed: 06/12/2023]
Abstract
The thermodynamics of the discrete nonlinear Schrödinger equation in the vicinity of infinite temperature is explicitly solved in the microcanonical ensemble by means of large-deviation techniques. A first-order phase transition between a thermalized phase and a condensed (localized) one occurs at the infinite-temperature line. Inequivalence between statistical ensembles characterizes the condensed phase, where the grand-canonical representation does not apply. The control over finite-size corrections of the microcanonical partition function allows to design an experimental test of delocalized negative-temperature states in lattices of cold atoms.
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Affiliation(s)
- Giacomo Gradenigo
- Gran Sasso Science Institute, Viale F. Crispi 7, 67100, L'Aquila, Italy.
| | - Stefano Iubini
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, 50019, Sesto Fiorentino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino, Italy
| | - Roberto Livi
- Consiglio Nazionale delle Ricerche, Istituto dei Sistemi Complessi, Via Madonna del Piano 10, 50019, Sesto Fiorentino, Italy
- Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino, Italy
- Dipartimento di Fisica e Astronomia and CSDC, Università di Firenze, Via G. Sansone 1, 50019, Sesto Fiorentino, Italy
| | - Satya N Majumdar
- LPTMS, CNRS, Université Paris-Sud, Université Paris-Saclay, 91405, Orsay, France
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9
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Leykam D, Smolina E, Maluckov A, Flach S, Smirnova DA. Probing Band Topology Using Modulational Instability. PHYSICAL REVIEW LETTERS 2021; 126:073901. [PMID: 33666481 DOI: 10.1103/physrevlett.126.073901] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/07/2020] [Accepted: 01/13/2021] [Indexed: 06/12/2023]
Abstract
We analyze the modulational instability of nonlinear Bloch waves in topological photonic lattices. In the initial phase of the instability development captured by the linear stability analysis, long wavelength instabilities and bifurcations of the nonlinear Bloch waves are sensitive to topological band inversions. At longer timescales, nonlinear wave mixing induces spreading of energy through the entire band and spontaneous creation of wave polarization singularities determined by the band Chern number. Our analytical and numerical results establish modulational instability as a tool to probe bulk topological invariants and create topologically nontrivial wave fields.
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Affiliation(s)
- Daniel Leykam
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Korea
- Basic Science Program, Korea University of Science and Technology, Daejeon 34113, Korea
| | - Ekaterina Smolina
- Institute of Applied Physics, Russian Academy of Science, Nizhny Novgorod 603950, Russia
| | - Aleksandra Maluckov
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Korea
- P* Group, Vinča Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34126, Korea
- Basic Science Program, Korea University of Science and Technology, Daejeon 34113, Korea
| | - Daria A Smirnova
- Institute of Applied Physics, Russian Academy of Science, Nizhny Novgorod 603950, Russia
- Nonlinear Physics Centre, Australian National University, Canberra, Australian Capital Territory 2601, Australia
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10
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Many Manda B, Senyange B, Skokos C. Chaotic wave-packet spreading in two-dimensional disordered nonlinear lattices. Phys Rev E 2020; 101:032206. [PMID: 32289935 DOI: 10.1103/physreve.101.032206] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/19/2019] [Accepted: 02/05/2020] [Indexed: 11/07/2022]
Abstract
We reveal the generic characteristics of wave-packet delocalization in two-dimensional nonlinear disordered lattices by performing extensive numerical simulations in two basic disordered models: the Klein-Gordon system and the discrete nonlinear Schrödinger equation. We find that in both models (a) the wave packet's second moment asymptotically evolves as t^{a_{m}} with a_{m}≈1/5 (1/3) for the weak (strong) chaos dynamical regime, in agreement with previous theoretical predictions [S. Flach, Chem. Phys. 375, 548 (2010)CMPHC20301-010410.1016/j.chemphys.2010.02.022]; (b) chaos persists, but its strength decreases in time t since the finite-time maximum Lyapunov exponent Λ decays as Λ∝t^{α_{Λ}}, with α_{Λ}≈-0.37 (-0.46) for the weak (strong) chaos case; and (c) the deviation vector distributions show the wandering of localized chaotic seeds in the lattice's excited part, which induces the wave packet's thermalization. We also propose a dimension-independent scaling between the wave packet's spreading and chaoticity, which allows the prediction of the obtained α_{Λ} values.
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Affiliation(s)
- B Many Manda
- Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - B Senyange
- Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
| | - Ch Skokos
- Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch, 7701 Cape Town, South Africa
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11
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Sun ZY, Yu X. Transient diffusion and two-regime localization of discrete breatherlike excitations in nonlinear Schrödinger lattice with disorder. Phys Rev E 2019; 100:022202. [PMID: 31574744 DOI: 10.1103/physreve.100.022202] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/20/2019] [Indexed: 11/07/2022]
Abstract
We systematically simulate and analyze the motion of discrete breatherlike excitations (DBEs) in the nonlinear Schrödinger lattice with random potentials. A universal transient diffusion of the DBEs is observed for short timescales (t≲10^{3}). For longer timescales (t up to 10^{5}), the DBEs become localized. Such localization, depending on the DBE powers, has two different regimes: Regime I is the Anderson-like localization induced by the disorder, while Regime II is an enhanced localization attributed to both the disorder and discreteness. Our study is expected to shed light on understanding the interplay between disorder and strong nonlinearity, from the diffusive transport and localization properties of nonlinear localized excitations in random media.
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Affiliation(s)
- Zhi-Yuan Sun
- Institute of Fluid Mechanics, Beihang University, Beijing 100191, China.,International Research Institute for Multidisciplinary Science, Beihang University, Beijing 100191, China
| | - Xin Yu
- Institute of Fluid Mechanics, Beihang University, Beijing 100191, China
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12
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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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13
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On the Wave Turbulence Theory for the Nonlinear Schrödinger Equation with Random Potentials. ENTROPY 2019. [PMCID: PMC7515352 DOI: 10.3390/e21090823] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
Abstract
We derive new kinetic and a porous medium equations from the nonlinear Schrödinger equation with random potentials. The kinetic equation has a very similar form compared to the four-wave turbulence kinetic equation in the wave turbulence theory. Moreover, we construct a class of self-similar solutions for the porous medium equation. These solutions spread with time, and this fact answers the “weak turbulence” question for the nonlinear Schrödinger equation with random potentials. We also derive Ohm’s law for the porous medium equation.
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14
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Chirondojan LF, Oppo GL. Multiple-time-scale analysis for pinned breathers in Bose-Hubbard chains. Phys Rev E 2019; 99:022212. [PMID: 30934338 DOI: 10.1103/physreve.99.022212] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/18/2018] [Indexed: 11/07/2022]
Abstract
Localized and pinned discrete breathers in Bose-Einstein condensates in optical lattices or in arrays of optical waveguides oscillate with frequencies which are much higher than those present in the spectrum of the background. Hence, the interaction between localized breathers and their surroundings is extremely weak leading to a multiple-time-scale perturbation expansion. We identify the leading order in the asymptotic expansion of the breather amplitude which does not average to zero after one full oscillation. The reduced model predicts a lower bound of the breather drift times and explains the topological differences between breathers in dimers, trimers, and in spatially extended one-dimensional lattices even in the presence of transport from boundary heat-baths. These analytical boundaries hold true for lattices of any length, due to the highly localized nature of breathers.
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Affiliation(s)
- Liviu F Chirondojan
- SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, EU
| | - Gian-Luca Oppo
- SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, EU
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15
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Iubini S, Chirondojan L, Oppo GL, Politi A, Politi P. Dynamical Freezing of Relaxation to Equilibrium. PHYSICAL REVIEW LETTERS 2019; 122:084102. [PMID: 30932580 DOI: 10.1103/physrevlett.122.084102] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/08/2018] [Revised: 01/23/2019] [Indexed: 06/09/2023]
Abstract
We provide evidence of an extremely slow thermalization occurring in the discrete nonlinear Schrödinger model. At variance with many similar processes encountered in statistical mechanics-typically ascribed to the presence of (free) energy barriers-here the slowness has a purely dynamical origin: it is due to the presence of an adiabatic invariant, which freezes the dynamics of a tall breather. Consequently, relaxation proceeds via rare events, where energy is suddenly released towards the background. We conjecture that this exponentially slow relaxation is a key ingredient contributing to the nonergodic behavior recently observed in the negative-temperature region of the discrete nonlinear Schrödinger equation.
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Affiliation(s)
- Stefano Iubini
- Department of Physics and Astronomy, University of Padova, Via Marzolo 8, I-35131 Padova, Italy
- Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy
| | - Liviu Chirondojan
- SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, United Kingdom
| | - Gian-Luca Oppo
- SUPA and Department of Physics, University of Strathclyde, Glasgow G4 0NG, Scotland, United Kingdom
| | - Antonio Politi
- Institute for Complex Systems and Mathematical Biology & SUPA University of Aberdeen, Aberdeen AB24 3UE, Scotland, United Kingdom
| | - Paolo Politi
- Istituto dei Sistemi Complessi, Consiglio Nazionale delle Ricerche, Via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy
- INFN Sezione di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy
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16
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Mithun T, Danieli C, Kati Y, Flach S. Dynamical Glass and Ergodization Times in Classical Josephson Junction Chains. PHYSICAL REVIEW LETTERS 2019; 122:054102. [PMID: 30822006 DOI: 10.1103/physrevlett.122.054102] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/08/2018] [Revised: 11/19/2018] [Indexed: 06/09/2023]
Abstract
Models of classical Josephson junction chains turn integrable in the limit of large energy densities or small Josephson energies. Close to these limits the Josephson coupling between the superconducting grains induces a short-range nonintegrable network. We compute distributions of finite-time averages of grain charges and extract the ergodization time T_{E} which controls their convergence to ergodic δ distributions. We relate T_{E} to the statistics of fluctuation times of the charges, which are dominated by fat tails. T_{E} is growing anomalously fast upon approaching the integrable limit, as compared to the Lyapunov time T_{Λ}-the inverse of the largest Lyapunov exponent-reaching astonishing ratios T_{E}/T_{Λ}≥10^{8}. The microscopic reason for the observed dynamical glass is rooted in a growing number of grains evolving over long times in a regular almost integrable fashion due to the low probability of resonant interactions with the nearest neighbors. We conjecture that the observed dynamical glass is a generic property of Josephson junction networks irrespective of their space dimensionality.
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Affiliation(s)
- Thudiyangal Mithun
- 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
| | - Yagmur Kati
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
- Basic Science Program, Korea University of Science and Technology (UST), Daejeon 34113, Republic of Korea
| | - Sergej Flach
- Center for Theoretical Physics of Complex Systems, Institute for Basic Science, Daejeon 34051, Korea
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17
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Aghion E, Kessler DA, Barkai E. From Non-Normalizable Boltzmann-Gibbs Statistics to Infinite-Ergodic Theory. PHYSICAL REVIEW LETTERS 2019; 122:010601. [PMID: 31012666 DOI: 10.1103/physrevlett.122.010601] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/17/2018] [Revised: 08/30/2018] [Indexed: 06/09/2023]
Abstract
We study a particle immersed in a heat bath, in the presence of an external force which decays at least as rapidly as 1/x, e.g., a particle interacting with a surface through a Lennard-Jones or a logarithmic potential. As time increases, our system approaches a non-normalizable Boltzmann state. We study observables, such as the energy, which are integrable with respect to this asymptotic thermal state, calculating both time and ensemble averages. We derive a useful canonical-like ensemble which is defined out of equilibrium, using a maximum entropy principle, where the constraints are normalization, finite averaged energy, and a mean-squared displacement which increases linearly with time. Our work merges infinite-ergodic theory with Boltzmann-Gibbs statistics, thus extending the scope of the latter while shedding new light on the concept of ergodicity.
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Affiliation(s)
- Erez Aghion
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
- Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - David A Kessler
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
| | - Eli Barkai
- Department of Physics, Bar-Ilan University, Ramat-Gan 52900, Israel
- Institute of Nanotechnology and Advanced Materials, Bar-Ilan University, Ramat-Gan 52900, Israel
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