Glassy dynamics in strongly anharmonic chains of oscillators

Wojciech De Roeck; François Huveneers

doi:10.1016/j.crhy.2019.08.007

Fourier and the science of today / Fourier et la science d'aujourd'hui

Glassy dynamics in strongly anharmonic chains of oscillators
[Dynamique vitreuse pour des chaînes d'oscillateurs fortement anharmoniques]

Wojciech De Roeck ¹ ; François Huveneers ²

¹ Instituut Theoretische Fysica, KU Leuven, 3001 Leuven, Belgium
² Ceremade, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, 75775 Paris cedex 16, France

Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 419-428.

Résumé
Abstract

On propose un état de l'art sur les mécanismes de transport dans les chaînes d'oscillateurs fortement anharmoniques, proches de la limite atomique où tous les oscillateurs sont découplés. Dans ce régime, le mouvement de la plupart des oscillateurs reste pratiquement intégrable, c'est-à-dire quasi périodique, sur des échelles de temps très longues, alors que la dynamique est chaotique en quelques rares endroits, qui bougent très lentement et conduisent à une très lente redistribution de l'énergie à travers le système. Le matériau acquiert plusieurs caractéristiques propres aux verres : intermittence, encombrement, et une réduction drastique de la mobilité en fonction des paramètres thermodynamiques. On considère principalement des systèmes classiques, mais également certains systèmes quantiques. On analyse aussi le lien avec des systèmes désordonnés, dont la physique est similaire dans une large mesure.

We review the mechanism for transport in strongly anharmonic chains of oscillators near the atomic limit where all oscillators are decoupled. In this regime, the motion of most oscillators remains close to integrable, i.e. quasi-periodic, on very long time scales, while a few chaotic spots move very slowly and redistribute the energy across the system. The material acquires several characteristic properties of dynamical glasses: intermittency, jamming, and a drastic reduction of the mobility as a function of the thermodynamical parameters. We consider both classical and quantum systems, though with more emphasis on the former, and we discuss also the connections with quenched disordered systems, which display a similar physics to a large extent.

Publié le : 2019-07-01

DOI : 10.1016/j.crhy.2019.08.007

Keywords: Glass, Thermal conductivity, Transport, Chaos, Localization
Mot clés : Verre, Conductivité thermique, Transport, Chaos, Localisation

Affiliations des auteurs :

Wojciech De Roeck ¹ ; François Huveneers ²

¹ Instituut Theoretische Fysica, KU Leuven, 3001 Leuven, Belgium
² Ceremade, UMR CNRS 7534, Université Paris-Dauphine, PSL Research University, 75775 Paris cedex 16, France

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     title = {Glassy dynamics in strongly anharmonic chains of oscillators},
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Wojciech De Roeck; François Huveneers. Glassy dynamics in strongly anharmonic chains of oscillators. Comptes Rendus. Physique, Volume 20 (2019) no. 5, pp. 419-428. doi : 10.1016/j.crhy.2019.08.007. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2019.08.007/

Bibliographie
Cité par

[1] M.S. Green Markoff random processes and the statistical mechanics of time-dependent phenomena. II. Irreversible processes in fluids, J. Chem. Phys., Volume 22 (1954) no. 3, pp. 398-413

[2] R. Kubo Statistical-mechanical theory of irreversible processes. I. General theory and simple applications to magnetic and conduction problems, J. Phys. Soc. Jpn., Volume 12 (1957) no. 6, pp. 570-586

[3] R. Kubo The fluctuation–dissipation theorem, Rep. Prog. Phys., Volume 29 (1966) no. 1, pp. 255-284

[4] R. Lefevere; A. Schenkel Normal heat conductivity in a strongly pinned chain of anharmonic oscillators, J. Stat. Mech. Theory Exp., Volume 2006 (2006) no. 02

[5] K. Aoki; J. Lukkarinen; H. Spohn Energy transport in weakly anharmonic chains, J. Stat. Phys., Volume 124 (2006) no. 5, pp. 1105-1129

[6] J. Lukkarinen Kinetic theory of phonons in weakly anharmonic particle chains (S. Lepri, ed.), Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer, Springer International Publishing, Cham, 2016, pp. 159-214

[7] D. Escande; H. Kantz; R. Livi; S. Ruffo Self-consistent check of the validity of Gibbs calculus using dynamical variables, J. Stat. Phys., Volume 76 (1994) no. 1–2, pp. 605-626

[8] V. Oganesyan; A. Pal; D. Huse Energy transport in disordered classical spin chains, Phys. Rev. B, Volume 80 (2009) no. 11

[9] D. Basko Weak chaos in the disordered nonlinear Schrödinger chain: destruction of Anderson localization by Arnold diffusion, Ann. Phys., Volume 326 (2011) no. 7, pp. 1577-1655

[10] W. De Roeck; F. Huveneers Asymptotic quantum many-body localization from thermal disorder, Commun. Math. Phys., Volume 332 (2014) no. 3, pp. 1017-1082

[11] W. De Roeck; F. Huveneers Asymptotic localization of energy in nondisordered oscillator chains, Commun. Pure Appl. Math., Volume 68 (2015) no. 9, pp. 1532-1568

[12] W. De Roeck; F. Huveneers Can translation invariant systems exhibit a many-body localized phase? (P. Gonçalves; A.J. Soares, eds.), From Particle Systems to Partial Differential Equations II: Particle Systems and PDEs II, Springer International Publishing, Braga, Portugal, December 2013 , pp. 173-192 (2015)

[13] W. De Roeck; F. Huveneers; M. Müller; M. Schiulaz Absence of many-body mobility edges, Phys. Rev. B, Volume 93 (2016)

[14] M. Pino; L.B. Ioffe; B.L. Altshuler Nonergodic metallic and insulating phases of Josephson junction chains, Proc. Natl. Acad. Sci. USA, Volume 113 (2016) no. 3, pp. 536-541

[15] F. Huveneers Classical and quantum systems: transport due to rare events, Ann. Phys., Volume 529 (2017) no. 7

[16] C. Danieli; T. Mithun; Y. Kati; D.K. Campbell; S. Flach Dynamical glass in weakly non-integrable many-body systems, 2018 | arXiv

[17] T. Mithun; C. Danieli; Y. Kati; S. Flach Dynamical glass and ergodization times in classical Josephson junction chains, Phys. Rev. Lett., Volume 122 (2019)

[18] J. Pöschel A lecture on the classical KAM-theorem, Proc. Symp. Pure Math., Volume 69 (2001), pp. 707-732

[19] J. Pöschel Nekhoroshev estimates for quasi-convex Hamiltonian systems, Math. Z., Volume 213 (1993) no. 1, pp. 187-216

[20] P.W. Anderson Absence of diffusion in certain random lattices, Phys. Rev., Volume 109 (1958), pp. 1492-1505

[21] L. Fleishman; P.W. Anderson Interactions and the Anderson transition, Phys. Rev. B, Volume 21 (1980) no. 6, p. 2366

[22] I. Gornyi; A. Mirlin; D. Polyakov Interacting electrons in disordered wires: Anderson localization and low-T transport, Phys. Rev. Lett., Volume 95 (2005) no. 20

[23] D.M. Basko; I.L. Aleiner; B.L. Altshuler Metal–insulator transition in a weakly interacting many-electron system with localized single-particle states, Ann. Phys., Volume 321 (2006) no. 5, pp. 1126-1205

[24] V. Oganesyan; D.A. Huse Localization of interacting fermions at high temperature, Phys. Rev. B, Volume 75 (2007)

[25] J. Imbrie On many-body localization for quantum spin chains, J. Stat. Phys., Volume 163 (2016) no. 5, pp. 998-1048

[26] J. Imbrie Diagonalization and many-body localization for a disordered quantum spin chain, Phys. Rev. Lett., Volume 117 (2016)

[27] K. Agarwal; S. Gopalakrishnan; M. Knap; M. Müller; E. Demler Anomalous diffusion and Griffiths effects near the many-body localization transition, Phys. Rev. Lett., Volume 114 (2015) no. 16

[28] S. Gopalakrishnan; K. Agarwal; E. Demler; D. Huse; M. Knap Griffiths effects and slow dynamics in nearly many-body localized systems, Phys. Rev. B, Volume 93 (2016) no. 13

[29] K. Agarwal; E. Altman; E. Demler; S. Gopalakrishnan; D.A. Huse; M. Knap Rare-region effects and dynamics near the many-body localization transition, Ann. Phys., Volume 529 (2017) no. 7

[30] B. Chirikov A universal instability of many-dimensional oscillator systems, Phys. Rep., Volume 52 (1979) no. 5, pp. 263-379

[31] F. Huveneers Energy transport through rare collisions, J. Stat. Phys., Volume 146 (2012) no. 1, pp. 73-97

[32] G. Benettin; J. Fröhlich; A. Giorgilli A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom, Commun. Math. Phys., Volume 119 (1988) no. 1, pp. 95-108

[33] G. Basile; C. Bernardin; S. Olla Momentum conserving model with anomalous thermal conductivity in low dimensional systems, Phys. Rev. Lett., Volume 96 (2006) no. 20

[34] B. Rumpf Simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities, Phys. Rev. E, Volume 69 (2004)

[35] S. Iubini; R. Franzosi; R. Livi; G.-L. Oppo; A. Politi Discrete breathers and negative-temperature states, New J. Phys., Volume 15 (2013) no. 2

[36] S. Chatterjee A note about the uniform distribution on the intersection of a simplex and a sphere, J. Topol. Anal., Volume 9 (2017) no. 4, pp. 717-738

[37] T. Mithun; Y. Kati; C. Danieli; S. Flach Weakly nonergodic dynamics in the Gross-Pitaevskii lattice, Phys. Rev. Lett., Volume 120 (2018)

[38] F. Huveneers; E. Theil Equivalence of ensembles, condensation and glassy dynamics in the Bose–Hubbard Hamiltonian, 2019 | arXiv

[39] M. Hairer; J.C. Mattingly Slow energy dissipation in anharmonic oscillator chains, Commun. Pure Appl. Math., Volume 62 (2009) no. 8, pp. 999-1032

[40] N. Cuneo; J.-P. Eckmann; C.E. Wayne Energy dissipation in Hamiltonian chains of rotators, Nonlinearity, Volume 30 (2017) no. 11, p. R81-R117

[41] N. Cuneo; J.-P. Eckmann; M. Hairer; L. Rey-Bellet Non-equilibrium steady states for networks of oscillators, Electron. J. Probab., Volume 23 (2018) (28 pp)

[42] N. Cuneo; J.-P. Eckmann; C. Poquet Non-equilibrium steady state and subgeometric ergodicity for a chain of three coupled rotors, Nonlinearity, Volume 28 (2015) no. 7, pp. 2397-2421

[43] N. Cuneo; J.-P. Eckmann Non-equilibrium steady states for chains of four rotors, Commun. Math. Phys., Volume 345 (2016) no. 1, pp. 185-221

[44] N. Cuneo; C. Poquet On the relaxation rate of short chains of rotors interacting with Langevin thermostats, Electron. Commun. Probab., Volume 22 (2017) (8 pp)

[45] C. Bernardin; F. Huveneers Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential, Probab. Theory Relat. Fields, Volume 157 (2013) no. 1, pp. 301-331

[46] W. De Roeck; A. Dhar; F. Huveneers; M. Schütz Step density profiles in localized chains, J. Stat. Phys., Volume 167 (2017) no. 5, pp. 1143-1163

[47] S. Lepri; R. Livi; A. Politi Thermal conduction in classical low-dimensional lattices, Phys. Rep., Volume 377 (2003) no. 1, pp. 1-80

[48] A. Dhar; J.L. Lebowitz Effect of phonon-phonon interactions on localization, Phys. Rev. Lett., Volume 100 (2008)

[49] A. Dhar; K. Saito Heat conduction in the disordered Fermi-Pasta-Ulam chain, Phys. Rev. E, Volume 78 (2008)

[50] M. Mulansky; K. Ahnert; A. Pikovsky; D. Shepelyansky Dynamical thermalization of disordered nonlinear lattices, Phys. Rev. E, Volume 80 (2009)

[51] F. Huveneers Drastic fall-off of the thermal conductivity for disordered lattices in the limit of weak anharmonic interactions, Nonlinearity, Volume 26 (2013) no. 3, p. 837

[52] S. Fishman; Y. Krivolapov; A. Soffer On the problem of dynamical localization in the nonlinear Schrödinger equation with a random potential, J. Stat. Phys., Volume 131 (2008) no. 5, pp. 843-865

[53] S. Fishman; Y. Krivolapov; A. Soffer Perturbation theory for the nonlinear Schrödinger equation with a random potential, Nonlinearity, Volume 22 (2009) no. 12, p. 2861

[54] S. Fishman; Y. Krivolapov; A. Soffer The nonlinear Schrödinger equation with a random potential: results and puzzles, Nonlinearity, Volume 25 (2012) no. 4, p. R53

[55] J. Fröhlich; T. Spencer; C. Wayne Localization in disordered, nonlinear dynamical systems, J. Stat. Phys., Volume 42 (1986) no. 3–4, pp. 247-274

[56] J. Pöschel Small divisors with spatial structure in infinite dimensional Hamiltonian systems, Commun. Math. Phys., Volume 127 (1990) no. 2, pp. 351-393

[57] W.-M. Wang; Z. Zhang Long time Anderson localization for the nonlinear random Schrödinger equation, J. Stat. Phys., Volume 134 (2009) no. 5–6, pp. 953-968

[58] J. Bourgain; W.-M. Wang Diffusion bound for a nonlinear Schrödinger equation, Mathematical Aspect of Nonlinear Dispersive Equations, Ann. Math. Stud., 2007, pp. 21-42

[59] A.S. Pikovsky; D.L. Shepelyansky Destruction of Anderson localization by a weak nonlinearity, Phys. Rev. Lett., Volume 100 (2008)

[60] S. Flach; D.O. Krimer; C. Skokos Universal spreading of wave packets in disordered nonlinear systems, Phys. Rev. Lett., Volume 102 (2009)

[61] M. Ivanchenko; T. Laptyeva; S. Flach Anderson localization or nonlinear waves: a matter of probability, Phys. Rev. Lett., Volume 107 (2011)

[62] T. Laptyeva; M. Ivanchenko; S. Flach Nonlinear lattice waves in heterogeneous media, J. Phys. A, Math. Theor., Volume 47 (2014) no. 49

[63] M. Mulansky; A. Pikovsky Spreading in disordered lattices with different nonlinearities, Europhys. Lett., Volume 90 (2010) no. 1

[64] M. Mulansky; A. Pikovsky Energy spreading in strongly nonlinear disordered lattices, New J. Phys., Volume 15 (2013) no. 5

[65] I. Vakulchyk; M.V. Fistul; S. Flach Wave packet spreading with disordered nonlinear discrete-time quantum walks, Phys. Rev. Lett., Volume 122 (2019)

[66] R. Nandkishore; D.A. Huse Many-body localization and thermalization in quantum statistical mechanics, Annu. Rev. Condens. Matter Phys., Volume 6 (2015), pp. 15-38

[67] D.A. Abanin; Z. Papić Recent progress in many-body localization, Ann. Phys., Volume 529 (2017) no. 7

[68] F. Alet; N. Laflorencie Many-body localization: an introduction and selected topics, C. R. Physique, Volume 19 (2018) no. 6, pp. 498-525

[69] Y. Kagan; L. Maksimov Localization in a system of interacting particles diffusing in a regular crystal, Zh. Èksp. Teor. Fiz., Volume 87 (1984), pp. 348-365

[70] G. Carleo; F. Becca; M. Schiró; M. Fabrizio Localization and glassy dynamics of many-body quantum systems, Sci. Rep., Volume 2 (2012)

[71] T. Grover; M. Fisher Quantum disentangled liquids, J. Stat. Mech. Theory Exp., Volume 2014 (2014) no. 10

[72] M. Schiulaz; M. Müller Ideal quantum glass transitions: many-body localization without quenched disorder, AIP Conf. Proc., Volume 1610 (2014), pp. 11-23

[73] M. Schiulaz; A. Silva; M. Müller Dynamics in many-body localized quantum systems without disorder, Phys. Rev. B, Volume 91 (2015) no. 18

[74] A. Bols; W. De Roeck Asymptotic localization in the Bose–Hubbard model, J. Math. Phys., Volume 59 (2018) no. 2

[75] N.Y. Yao; C.R. Laumann; J.I. Cirac; M.D. Lukin; J.-E. Moore Quasi-many-body localization in translation-invariant systems, Phys. Rev. Lett., Volume 117 (2016)

[76] J. Hickey; S. Genway; J. Garrahan Signatures of many-body localisation in a system without disorder and the relation to a glass transition, J. Stat. Mech. Theory Exp., Volume 2016 (2016) no. 5

[77] A.A. Michailidis; M. Žnidarič; M. Medvedyeva; D.A. Abanin; T. Prosen; Z. Papić Slow dynamics in translation-invariant quantum lattice models, Phys. Rev. B, Volume 97 (2018)

[78] W. De Roeck; F. Huveneers Scenario for delocalization in translation-invariant systems, Phys. Rev. B, Volume 90 (2014)

[79] Z. Papić; E. Stoudenmire; D. Abanin Many-body localization in disorder-free systems: the importance of finite-size constraints, Ann. Phys., Volume 362 (2015), pp. 714-725

[80] W. De Roeck; F. Huveneers Stability and instability towards delocalization in many-body localization systems, Phys. Rev. B, Volume 95 (2017)

[81] D.J. Luitz; F. Huveneers; W. De Roeck How a small quantum bath can thermalize long localized chains, Phys. Rev. Lett., Volume 119 (2017)

[82] T. Thiery; F. Huveneers; M. Müller; W. De Roeck Many-body delocalization as a quantum avalanche, Phys. Rev. Lett., Volume 121 (2018)

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