Wave turbulence and thermalization in one-dimensional chains. (English) Zbl 07771889
Summary: One-dimensional chains are used as a fundamental model of condensed matter, and have constituted the starting point for key developments in nonlinear physics and complex systems. The pioneering work in this field was proposed by Fermi, Pasta, Ulam and Tsingou in the 50s in Los Alamos. An intense and fruitful mathematical and physical research followed during these last 70 years. Recently, a fresh look at the mechanisms at the route of thermalization of such systems has been provided through the lens of the Wave Turbulence approach. In this review, we give a critical summary of the results obtained in this framework. We also present a series of open problems and challenges that future work needs to address.
MSC:
| 82D03 | Statistical mechanics in condensed matter (general) |
Keywords:
one-dimensional chains; thermalization; wave turbulence; Fermi-Pasta-Ulam-Tsingou problem; wave kinetic equationReferences:
| [1] | Fermi, E.; Pasta, J.; Ulam, S., Los alamos report la-1940. Fermi, Collect. Pap., 977-988 (1955) |
| [2] | Poincaré, H., Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 2 (1893), Gauthier-Villars it fils · JFM 25.1847.03 |
| [3] | Ma, S.-K., Statistical Mechanics (1985), World Scientific Publishing Company · Zbl 0952.82500 |
| [4] | Castiglione, P.; Falcioni, M.; Lesne, A.; Vulpiani, A., Chaos and coarse graining in statistical mechanics. Chaos Coarse Graining Stat. Mech. (2008) · Zbl 1189.82001 |
| [5] | Chibbaro, S.; Rondoni, L.; Vulpiani, A., Reductionism, Emergence and Levels of Reality (2014), Springer |
| [6] | Fermi, E., Dimostrazione che in generale un sistema meccanico normale è quasi ergodico. Il Nuovo Cimento (1911-1923), 1, 267-269 (1923) |
| [7] | Gallavotti, G., The Fermi-Pasta-Ulam Problem: A Status Report, Vol. 728 (2007), Springer |
| [8] | Benenti, G.; Lepri, S.; Livi, R., Anomalous heat transport in classical many-body systems: Overview and perspectives. Front. Phys., 292 (2020) |
| [9] | Dauxois, T., Fermi, Pasta, Ulam and a mysterious lady. Phys. Today, 61, 1, 55-57 (2008) |
| [10] | Fermi, E.; Pasta, J.; Ulam, S., Studies of Nonlinear ProblemsTech. rep. (1955), I, Los Alamos Scientific Laboratory Report No. LA-1940 |
| [11] | The collected papers of enrico Fermi, vol. II: US 1939-1954) (1967), Chicago University Press |
| [12] | Zabusky, N. J.; Kruskal, M. D., Interaction of “solitons” in a collisionless plasma and the recurrence of initial states. Phys. Rev. Lett., 6, 240 (1965) · Zbl 1201.35174 |
| [13] | Zaslavski, G.; Chirikov, B., Stochastic instability of non-linear oscillations. Sov. Phys. Uspekhi, 5, 549 (1972) · Zbl 1156.34335 |
| [14] | Campbell, D. K.; Rosenau, P.; Zaslavsky, G. M., Introduction: the Fermi-Pasta-Ulam problem: the first fifty years. Chaos, 1 (2005) |
| [15] | S.C.L. M. Peyrard, D. Angelov, Fluctuations in the DNA double helix, Eur. Phys. J. Spec. Top. 147, 173-189. |
| [16] | Sonone, R. L.; Jain, S. R., Enumeration and stability analysis of simple periodic orbits in \(\beta \)-Fermi pasta Ulam lattice. AIP Conf. Proc. (2014) |
| [17] | Berezin, A. A., Fermi-pasta-Ulam spectrum raying for NDE purposes, 311-320 |
| [18] | Ponno, A.; Christodoulidi, H.; Skokos, C.; Flach, S., The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior. Chaos, 4 (2011), arXiv:1107.2626 |
| [19] | Berman, G. P.; Izrailev, F. M., The Fermi-Pasta-Ulam problem: fifty years of progress. Chaos (Woodbury, NY), 1, 15104 (2005) · Zbl 1080.37077 |
| [20] | Dauxois, T.; Khomeriki, R.; Piazza, F.; Ruffo, S., The anti-FPU problem. Chaos, 1 (2005) · Zbl 1080.37078 |
| [21] | Fishman, S.; Krivolapov, Y.; Soffer, A., The nonlinear Schrödinger equation with a random potential: results and puzzles. Nonlinearity, 4, R53 (2012) · Zbl 1236.35166 |
| [22] | Nazarenko, S.; Soffer, A.; Tran, M.-B., On the wave turbulence theory for the nonlinear Schrödinger equation with random potentials. Entropy, 9, 823 (2019) |
| [23] | Chang, C.-W.; Okawa, D.; Garcia, H.; Majumdar, A.; Zettl, A., Breakdown of Fourier’s law in nanotube thermal conductors. Phys. Rev. Lett., 7 (2008) |
| [24] | Chen, J.; Zhang, G.; Li, B., Remarkable reduction of thermal conductivity in silicon nanotubes. Nano Lett., 10, 3978-3983 (2010) |
| [25] | Shen, S.; Henry, A.; Tong, J.; Zheng, R.; Chen, G., Polyethylene nanofibres with very high thermal conductivities. Nature Nanotechnol., 4, 251 (2010) |
| [26] | Yang, N.; Zhang, G.; Li, B., Violation of Fourier’s law and anomalous heat diffusion in silicon nanowires. Nano Today, 2, 85-90 (2010) |
| [27] | Liu, S.; Xu, X.; Xie, R.; Zhang, G.; Li, B., Anomalous heat conduction and anomalous diffusion in low dimensional nanoscale systems. Eur. Phys. J. B, 10, 337 (2012) |
| [28] | Chen, G., Non-Fourier phonon heat conduction at the microscale and nanoscale. Nat. Rev. Phys., 8, 555-569 (2021) |
| [29] | Huberman, S.; Duncan, R. A.; Chen, K.; Song, B.; Chiloyan, V.; Ding, Z.; Maznev, A. A.; Chen, G.; Nelson, K. A., Observation of second sound in graphite at temperatures above 100 k. Science, 6438, 375-379 (2019) |
| [30] | Barbalinardo, G.; Chen, Z.; Dong, H.; Fan, Z.; Donadio, D., Ultrahigh convergent thermal conductivity of carbon nanotubes from comprehensive atomistic modeling. Phys. Rev. Lett., 2 (2021) |
| [31] | Lepri, S.; Livi, R.; Politi, A., On the anomalous thermal conductivity of one-dimensional lattices. Europhys. Lett., 3, 271 (1998) |
| [32] | Prosen, T.; Campbell, D. K., Momentum conservation implies anomalous energy transport in 1D classical lattices. Phys. Rev. Lett., 13, 2857 (2000) |
| [33] | Lepri, S.; Livi, R.; Politi, A., Thermal conduction in classical low-dimensional lattices. Phys. Rep., 1, 1-80 (2003) |
| [34] | Basile, G.; Bernardin, C.; Olla, S., Momentum conserving model with anomalous thermal conductivity in low dimensional systems. Phys. Rev. Lett., 20 (2006) |
| [35] | Dhar, A., Heat transport in low-dimensional systems. Adv. Phys., 5, 457-537 (2008) |
| [36] | Wang, J.; Dmitriev, S. V.; Xiong, D., Thermal transport in long-range interacting Fermi-Pasta-Ulam chains. Phys. Rev. Res., 1 (2020) |
| [37] | Lepri, S., Thermal Transport in Low Dimensions: From Statistical Physics to Nanoscale Heat Transfer, Vol. 921 (2016), Springer |
| [38] | Benettin, G.; Livi, R.; Ponno, A., The Fermi-Pasta-Ulam problem: scaling laws vs. initial conditions. J. Stat. Phys., 5, 873-893 (2009) · Zbl 1375.82058 |
| [39] | Benettin, G.; Carati, A.; Galgani, L.; Giorgilli, A., The Fermi pasta Ulam problem and the metastability perspective, 151-189 · Zbl 1151.82003 |
| [40] | Benettin, G.; Ponno, A., Time-Scales to Equipartition in the Fermi-Pasta-Ulam Problem: Finite-Size Effects and Thermodynamic Limit. J. Stat. Phys., 4, 793-812 (2011) · Zbl 1227.82008 |
| [41] | Pistone, L.; Chibbaro, S.; Bustamante, M. D.; Lvov, Y. V.; Onorato, M., Universal route to thermalization in weakly-nonlinear one-dimensional chains. Math. Eng., 01-04-672, 672 (2019) · Zbl 1435.81125 |
| [42] | Izrailev, F. M.; Chirikov, B. V., Statistical properties of a nonlinear string, 30-32 · Zbl 0149.23207 |
| [43] | Livi, R.; Pettini, M.; Ruffo, S.; Sparpaglione, M.; Vulpiani, A., Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model. Phys. Rev. A, 2, 1039 (1985) |
| [44] | Casetti, L.; Cerruti-Sola, M.; Pettini, M.; Cohen, E. G.D., The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems. Phys. Rev. E, 6, 6566 (1997) |
| [45] | DeLuca, J.; Lichtenberg, A. J.; Ruffo, S., Energy transitions and time scales to equipartition in the Fermi-Pasta-Ulam oscillator chain. Phys. Rev. E, 4, 2877 (1995) |
| [46] | Benettin, G.; Christodoulidi, H.; Ponno, A., The Fermi-Pasta-Ulam Problem and Its Underlying Integrable Dynamics. J. Stat. Phys., 1-18 (2013) |
| [47] | Benettin, G.; Ponno, A., Understanding the FPU state in FPU-like models. Math. Eng., 3 (2020) |
| [48] | Grava, T.; Maspero, A.; Mazzuca, G.; Ponno, A., Adiabatic invariants for the FPUT and Toda chain in the thermodynamic limit. Comm. Math. Phys., 2, 811-851 (2020) · Zbl 1462.37082 |
| [49] | Flach, S.; Ivanchenko, M.; Kanakov, O., Q-breathers and the Fermi-Pasta-Ulam problem. Phys. Rev. Lett., 6 (2005) |
| [50] | Gershgorin, B.; Lvov, Y. V.; Cai, D., Renormalized waves and discrete breathers in \(\beta \)-Fermi-Pasta-Ulam chains. Phys. Rev. Lett., 26 (2005) |
| [51] | Danieli, C.; Campbell, D.; Flach, S., Intermittent many-body dynamics at equilibrium. Phys. Rev. E, 6 (2017) |
| [52] | Christodoulidi, H.; Efthymiopoulos, C.; Bountis, T., Energy localization on q-tori, long-term stability, and the interpretation of Fermi-Pasta-Ulam recurrences. Phys. Rev. E, 1 (2010) |
| [53] | Zakharov, V. E.; L’vov, V. S.; Falkovich, G., Kolmogorov Spectra of Turbulence I: Wave Turbulence (2012), Springer Science & Business Media · Zbl 0786.76002 |
| [54] | Nazarenko, S., Wave Turbulence, Vol. 825 (2011), Springer Science & Business Media · Zbl 1220.76006 |
| [55] | Newell, A. C.; Rumpf, B., Wave turbulence. Annu. Rev. Fluid Mech., 59-78 (2011) · Zbl 1299.76006 |
| [56] | Galtier, S., Physics of Wave Turbulence (2022), Cambridge University Press |
| [57] | Onorato, M.; Vozella, L.; Proment, D.; Lvov, Y. V., A route to thermalization in the \(\alpha \)-Fermi-Pasta-Ulam system. Proc. Natl. Acad. Sci., 14, 4208-4213 (2015) |
| [58] | Lvov, Y. V.; Onorato, M., Double scaling in the relaxation time in the \(\beta \)-Fermi-Pasta-Ulam-Tsingou model. Phys. Rev. Lett., 14 (2018) |
| [59] | Pistone, L.; Onorato, M.; Chibbaro, S., Thermalization in the discrete nonlinear Klein-Gordon chain in the wave-turbulence framework. Europhys. Lett., 4, 44003 (2018) |
| [60] | Fu, W.; Zhang, Y.; Zhao, H., Universal law of thermalization for one-dimensional perturbed toda lattices. New J. Phys., 4 (2019) |
| [61] | Fu, W.; Zhang, Y.; Zhao, H., Universal scaling of the thermalization time in one-dimensional lattices. Phys. Rev. E, 1 (2019) |
| [62] | Bustamante, M. D.; Hutchinson, K.; Lvov, Y. V.; Onorato, M., Exact discrete resonances in the Fermi-Pasta-Ulam-Tsingou system. Commun. Nonlinear Sci. Numer. Simul., 437-471 (2019) · Zbl 1464.82012 |
| [63] | Pereverzev, A., Fermi-Pasta-Ulam \(\beta\) lattice: Peierls equation and anomalous heat conductivity. Phys. Rev. E, 5 (2003) |
| [64] | Aoki, K.; Lukkarinen, J.; Spohn, H., Energy transport in weakly anharmonic chains. J. Stat. Phys., 5, 1105-1129 (2006) · Zbl 1135.82326 |
| [65] | Mellet, A.; Merino-Aceituno, S., Anomalous energy transport in FPU-\( \beta\) chain. J. Stat. Phys., 3, 583-621 (2015) · Zbl 1360.82082 |
| [66] | Dematteis, G.; Rondoni, L.; Proment, D.; De Vita, F.; Onorato, M., Coexistence of ballistic and Fourier regimes in the \(\beta\) Fermi-Pasta-Ulam-Tsingou lattice. Phys. Rev. Lett. (2020) |
| [67] | De Vita, F.; Dematteis, G.; Mazzilli, R.; Proment, D.; Lvov, Y. V.; Onorato, M., Anomalous conduction in one-dimensional particle lattices: Wave-turbulence approach. Phys. Rev. E, 3 (2022) |
| [68] | Zaleski, J.; Onorato, M.; Lvov, Y. V., Anomalous correlators in nonlinear dispersive wave systems. Phys. Rev. X, 2 (2020) |
| [69] | Picozzi, A.; Garnier, J.; Hansson, T.; Suret, P.; Randoux, S.; Millot, G.; Christodoulides, D. N., Optical wave turbulence: Towards a unified nonequilibrium thermodynamic formulation of statistical nonlinear optics. Phys. Rep., 1, 1-132 (2014) |
| [70] | Spohn, H., The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics.. J. Stat. Phys. (2006) · Zbl 1106.82033 |
| [71] | Lukkarinen, J., Kinetic theory of phonons in weakly anharmonic particle chains, 159-214 |
| [72] | Buckmaster, T.; Germain, P.; Hani, Z.; Shatah, J., On the kinetic wave turbulence description for NLS. Quart. Appl. Math., 2, 261-275 (2020) · Zbl 1431.35164 |
| [73] | Staffilani, G.; Tran, M.-B., On the wave turbulence theory for stochastic and random multidimensional KdV type equations (2021), arXiv preprint arXiv:2106.09819 |
| [74] | Deng, Y.; Hani, Z., On the derivation of the wave kinetic equation for NLS · Zbl 1479.35771 |
| [75] | Deng, Y.; Hani, Z., Full derivation of the wave kinetic equation (2021), arXiv preprint arXiv:2104.11204 |
| [76] | Ford, J., The Fermi-Pasta-Ulam problem: paradox turns discovery. Phys. Rep., 271 (1992) |
| [77] | Weissert, T. P., The Genesis of Simulation in Dynamics: Pursuing the Fermi-Pasta-Ulam Problem (1999) · Zbl 0908.58068 |
| [78] | Berman, G. P.; Izrailev, F. M., The Fermi-Pasta-Ulam problem: 50 years of progress. Chaos (2005) · Zbl 1080.37077 |
| [79] | Carati, A.; Galgani, L.; Giorgilli, A., The Fermi-Pasta-Ulam problem as a challenge for the foundations of physics. Chaos, 1 (2005) · Zbl 1080.82002 |
| [80] | Gallavotti, G., The Fermi-Pasta-Ulam Problem: A Status Report, Vol. 728 (2008), Springer · Zbl 1138.81004 |
| [81] | Kaufman, A. N., Wave entropy: A derivation by jaynes’ principle. Phys. Fluids, 7, 2326 (1986) |
| [82] | Makris, K. G.; Wu, F. O.; Jung, P. S.; Christodoulides, D. N., Statistical mechanics of weakly nonlinear optical multimode gases. Opt. Lett., 7, 1651-1654 (2020) |
| [83] | Rumpf, B., Transition behavior of the discrete nonlinear Schrödinger equation. Phys. Rev. E, 3 (2008) |
| [84] | Rumpf, B., Stable and metastable states and the formation and destruction of breathers in the discrete nonlinear Schrödinger equation. Physica D, 20, 2067-2077 (2009) · Zbl 1194.34012 |
| [85] | Rumpf, B., Growth and erosion of a discrete breather interacting with Rayleigh-Jeans distributed phonons. Europhys. Lett., 2, 26001 (2007) |
| [86] | Baldovin, M.; Iubini, S.; Livi, R.; Vulpiani, A., Statistical mechanics of systems with negative temperature. Phys. Rep., 1-50 (2021) · Zbl 1512.82019 |
| [87] | Onorato, M.; Dematteis, G.; Proment, D.; Pezzi, A.; Ballarin, M.; Rondoni, L., Equilibrium and nonequilibrium description of negative temperature states in a one-dimensional lattice using a wave kinetic approach. Phys. Rev. E, 1 (2022) |
| [88] | Dauxois, T.; Peyrard, M., Physics of Solitons (2006), Cambridge University Press · Zbl 1192.35001 |
| [89] | Ablowitz, M. J., Nonlinear Dispersive Waves: Asymptotic Analysis and Solitons, Vol. 47 (2011), Cambridge University Press · Zbl 1232.35002 |
| [90] | Faddeev, L. D., The inverse problem in the quantum theory of scattering. Uspekhi Mat. Nauk, 4, 57-119 (1959) · Zbl 0091.21902 |
| [91] | Faddeyev, L. D.; Seckler, B., The inverse problem in the quantum theory of scattering. J. Math. Phys., 1, 72-104 (1963) · Zbl 0112.45101 |
| [92] | Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving the Korteweg-deVries equation. Phys. Rev. Lett., 19, 1095 (1967) · Zbl 1103.35360 |
| [93] | Lax, P. D., Integrals of nonlinear equations of evolution and solitary waves. Commun. Pure Appl. Math., 5, 467-490 (1968) · Zbl 0162.41103 |
| [94] | Zakharov, V. E.; Faddeev, L. D., Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funktsional’nyi Analiz i ego Prilozheniya, 4, 18-27 (1971) |
| [95] | Shabat, A.; Zakharov, V., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys.—JETP, 1, 62 (1972) |
| [96] | Zakharov, V. E.; Shabat, A. B., A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. I. Funct. Anal. Appl., 3, 226-235 (1974) · Zbl 0303.35024 |
| [97] | Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Nonlinear-evolution equations of physical significance. Phys. Rev. Lett., 2, 125 (1973) · Zbl 1243.35143 |
| [98] | Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., Method for solving the sine-Gordon equation. Phys. Rev. Lett., 25, 1262 (1973) |
| [99] | Ponno, A., Soliton theory and the Fermi-Pasta-Ulam problem in the thermodynamic limit. Europhys. Lett., 5, 606 (2003) |
| [100] | Bambusi, D.; Ponno, A., On metastability in FPU. Comm. Math. Phys., 2, 539-561 (2006) · Zbl 1233.37049 |
| [101] | Gallone, M.; Ponno, A.; Rink, B., Korteweg-de Vries and Fermi-Pasta-Ulam-Tsingou: asymptotic integrability of quasi unidirectional waves. J. Phys. A, 30 (2021) · Zbl 1519.37080 |
| [102] | Infeld, E.; Rowlands, G., Nonlinear Waves, Solitons and Chaos (2000), Cambridge University Press · Zbl 0726.76018 |
| [103] | Ashcroft, N. W.; Mermin, N. D., Solid State Physics (2022), Cengage Learning · Zbl 1118.82001 |
| [104] | Davydov, A. S., The theory of contraction of proteins under their excitation. J. Theoret. Biol., 3, 559-569 (1973) |
| [105] | Putnam, B.; Van Zandt, L.; Prohofsky, E.; Mei, W., Resonant and localized breathing modes in terminal regions of the DNA double helix. Biophys. J., 2, 271-287 (1981) |
| [106] | Peyrard, M.; Bishop, A. R., Statistical mechanics of a nonlinear model for DNA denaturation. Phys. Rev. Lett., 23, 2755 (1989) |
| [107] | Gariaev, P. P.; Chudin, V. I.; Komissarov, G. G.; Berezin, A. A.; Vasiliev, A. A., Holographic associative memory of biological systems, 280-291 |
| [108] | Guasoni, M.; Garnier, J.; Rumpf, B.; Sugny, D.; Fatome, J.; Amrani, F.; Millot, G.; Picozzi, A., Incoherent Fermi-Pasta-Ulam recurrences and unconstrained thermalization mediated by strong phase correlations. Phys. Rev. X, 1 (2017) |
| [109] | Trillo, S.; Deng, G.; Biondini, G.; Klein, M.; Clauss, G.; Chabchoub, A.; Onorato, M., Experimental observation and theoretical description of multisoliton fission in shallow water. Phys. Rev. Lett., 14 (2016) |
| [110] | Driscoll, C.; O’Neil, T., Explanation of instabilities observed on a Fermi-Pasta-Ulam lattice. Phys. Rev. Lett., 2, 69 (1976) |
| [111] | Wadati, M., The modified Korteweg-de Vries equation. J. Phys. Soc. Japan, 5, 1289-1296 (1973) · Zbl 1334.35299 |
| [112] | Pace, S. D.; Reiss, K. A.; Campbell, D. K., The \(\beta\) Fermi-Pasta-Ulam-Tsingou recurrence problem. Chaos, 11 (2019) · Zbl 1432.37099 |
| [113] | Berman, G.; Kolovsky, A., The limit of stochasticity for a one-dimensional chain of interacting oscillators. Zh. Eksp. Teor. Fiz, 1938 (1984) |
| [114] | Ablowitz, M. J.; Kaup, D. J.; Newell, A. C.; Segur, H., The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math., 4, 249-315 (1974) · Zbl 0408.35068 |
| [115] | Flach, S.; Gorbach, A. V., Discrete breathers - advances in theory and applications. Phys. Rep., 1-3, 1-116 (2008) |
| [116] | Yuen, H. C.; Ferguson Jr., W. E., Relationship between benjamin-feir instability and recurrence in the nonlinear Schrödinger equation. Phys. Fluids, 8, 1275-1278 (1978) |
| [117] | Sulem, C.; Sulem, P.-L., The Nonlinear Schrödinger Equation: Self-Focusing and Wave Collapse, Vol. 139 (2007), Springer Science & Business Media |
| [118] | Zakharov, V.; Ostrovsky, L., Modulation instability: the beginning. Physica D, 5, 540-548 (2009) · Zbl 1157.37337 |
| [119] | Kuznetsov, E. A., Fermi-Pasta-Ulam recurrence and modulation instability. JETP Lett., 2, 125-129 (2017) |
| [120] | Ponno, A.; Christodoulidi, H.; Skokos, C.; Flach, S., The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior. Chaos, 4 (2011) |
| [121] | Toda, M., Vibration of a chain with nonlinear interaction. J. Phys. Soc. Japan, 2, 431-436 (1967) |
| [122] | Flaschka, H., The Toda lattice. II. Existence of integrals. Phys. Rev. B, 4, 1924 (1974) · Zbl 0942.37504 |
| [123] | Flaschka, H., On the Toda lattice. II: inverse-scattering solution. Progr. Theoret. Phys., 3, 703-716 (1974) · Zbl 0942.37505 |
| [124] | Benettin, G.; Ponno, A., FPU model and Toda model: A survey, A view. Kinetic Theory Turbul. Model. (2023) |
| [125] | Goldfriend, T.; Kurchan, J., Equilibration of quasi-integrable systems. Phys. Rev. E, 2 (2019) |
| [126] | Arnold, V. I.; Kozlov, V. V.; Neishtadt, A. I.; Iacob, I., Mathematical Aspects of Classical and Celestial Mechanics, Vol. 3 (2006), Springer · Zbl 1105.70002 |
| [127] | Giorgilli, A., Notes on Hamiltonian Dynamical Systems, Vol. 102 (2022), Cambridge University Press · Zbl 1517.37001 |
| [128] | Kolmogorov, A. N., On conservation of conditionally periodic motions for a small change in hamilton’s function, 527-530 · Zbl 0056.31502 |
| [129] | Arnold, V., Proof of a theorem of AN Kolmogorov on the conservation of quasiperiodic motions under a small change of the Hamiltonian function. Russ. Math. Surv, 5, 9-36 (1963) |
| [130] | Möser, J., On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen, II, 1-20 (1962) · Zbl 0107.29301 |
| [131] | Pöschel, J., A lecture on the classical KAM theorem (2009), arXiv preprint arXiv:0908.2234 |
| [132] | Benettin, G.; Galgani, L.; Giorgilli, A.; Strelcyn, J., Proof of Kolmogorov’s theorem on invariant tori using canonical transformations defined by the Lie method. Nuovo Cimento B;(Italy), 2 (1984) |
| [133] | Pöschel, J., Integrability of Hamiltonian systems on Cantor sets (1982), ETH Zurich, (Ph.D. thesis) · Zbl 0542.58015 |
| [134] | Chierchia, L.; Gallavotti, G., Smooth prime integrals for quasi-integrable Hamiltonian systems. Il Nuovo Cimento B (1971-1996), 2, 277-295 (1982) |
| [135] | Gallavotti, G., La meccanica classica e la rivoluzione quantistica nei lavori giovanili di Fermi. C. Bernardini e L. Bonolis (a cura di), Conoscere Fermi, Societ{à} Italiana di Fisica, Editrice Compositori, Bologna, 76 (2001) |
| [136] | Nekhoroshev, N. N., Behavior of Hamiltonian systems close to integrable. Funktsional’nyi Analiz i ego Prilozheniya, 4, 82-83 (1971) |
| [137] | Nekhoroshev, N. N., An exponential estimate of the time of stability of nearly-integrable Hamiltonian systems. Russian Math. Surveys, 6, 1 (1977) · Zbl 0389.70028 |
| [138] | Nekhoroshev, N., An exponential estimate of the stability time of nearly integrable Hamiltonian systems, vol II. Trudy Sem Imeni. in: Petrovskogo IG, 5 (1979) |
| [139] | Lochak, P.; Neishtadt, A., Estimates of stability time for nearly integrable systems with a quasiconvex Hamiltonian. Chaos, 4, 495-499 (1992) · Zbl 1055.37573 |
| [140] | Pöschel, J., Nekhoroshev estimates for quasi-convex Hamiltonian systems. Math. Z, 2, 187-216 (1993) · Zbl 0857.70009 |
| [141] | Guzzo, M.; Chierchia, L.; Benettin, G., The steep Nekhoroshev’s theorem. Comm. Math. Phys., 569-601 (2016) · Zbl 1337.37038 |
| [142] | Morbidelli, A.; Giorgilli, A., On a connection between KAM and Nekhoroshev’s theorems. Physica D, 3, 514-516 (1995) · Zbl 0890.58077 |
| [143] | Birkhoff, G. D., Dynamical Systems, Vol. 9 (1927), American Mathematical Soc. · JFM 53.0732.01 |
| [144] | Bambusi, D., An Introduction to Birkhoff Normal Form (2014), Universita di Milano |
| [145] | Rink, B., Fermi Pasta Ulam systems (FPU): mathematical aspects. Scholarpedia, 12, 9217 (2009) |
| [146] | Nishida, T., A note on an existence of conditionally periodic oscillation in a one-dimensional anharmonic lattice. Mem. Fac. Engrg. Kyoto Univ., 27-34 (1971) |
| [147] | Rink, B., Proof of Nishida’s conjecture on anharmonic lattices. Comm. Math. Phys., 3, 613-627 (2006) · Zbl 1113.82045 |
| [148] | Rink, B., Symmetry and resonance in periodic FPU chains. Comm. Math. Phys., 3, 665-685 (2001) · Zbl 0997.37057 |
| [149] | Rink, B., Direction-reversing traveling waves in the even Fermi-Pasta-Ulam lattice. J. Nonlinear Sci., 5 (2002) · Zbl 1012.37051 |
| [150] | Henrici, A.; Kappeler, T., Results on normal forms for FPU chains. Comm. Math. Phys., 1, 145-177 (2008) · Zbl 1168.70005 |
| [151] | Henrici, A., Normal Form for Fermi-Pasta-Ulam Chains (2008), University of Zurich, (Ph.D. thesis) |
| [152] | Henrici, A.; Kappeler, T., Nekhoroshev theorem for the periodic Toda lattice. Chaos, 3 (2009) · Zbl 1317.37057 |
| [153] | Kappeler, T.; Henrici, A., Resonant normal form for even periodic FPU chains. J. Eur. Math. Soc., 5, 1025-1056 (2009) · Zbl 1181.37081 |
| [154] | Ferguson, Jr., W.; Flaschka, H.; McLaughlin, D., Nonlinear normal modes for the Toda chain. J. Comput. Phys., 2, 157-209 (1982) · Zbl 0557.70028 |
| [155] | Bambusi, D.; Carati, A.; Maiocchi, A.; Maspero, A., Some analytic results on the FPU paradox, 235-254 · Zbl 1339.37074 |
| [156] | Bambusi, D.; Maspero, A., Birkhoff coordinates for the Toda lattice in the limit of infinitely many particles with an application to FPU. J. Funct. Anal., 5, 1818-1887 (2016) · Zbl 1335.37047 |
| [157] | Onorato, M.; Vozella, L.; Proment, D.; Lvov, Y. V., Route to thermalization in the \(\alpha \)-Fermi-Pasta-Ulam system. Proc. Natl. Acad. Sci., 14, 4208-4213 (2015) |
| [158] | Arnold, V. I.; Kozlov, V. V.; Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, Vol. 3 (2007), Springer Science & Business Media |
| [159] | Ganapa, S., Quasiperiodicity in the \(\alpha \)- Fermi-Pasta-Ulam-Tsingou system revisited: An approach using ideas from wave turbulence (2023), arXiv preprint arXiv:2303.10297 |
| [160] | Pistone, L.; Chibbaro, S.; Bustamante, M.; Lvov, Y.; Onorato, M., Universal route to thermalization in weakly-nonlinear one-dimensional chains. Math. Eng., 4, 672-698 (2019) · Zbl 1435.81125 |
| [161] | Lvov, V. S., Wave Turbulence Under Parametric Excitations, Applications to Magnets (1994), Springer-Verlag · Zbl 0814.76002 |
| [162] | Peierls, R., On the kinetic theory of thermal conduction in crystals. Ann. Phys., 1055 (1929) · JFM 55.0547.01 |
| [163] | Zakharov, V. E.; Filonenko, N., Energy spectrum for stochastic oscillations of the surface of a liquid. Dokl. Akad. Nauk, 6, 1292-1295 (1966) |
| [164] | Eyink, G. L.; Shi, Y.-K., Kinetic wave turbulence. Physica D, 18, 1487-1511 (2012) · Zbl 1287.82022 |
| [165] | Chibbaro, S.; Dematteis, G.; Rondoni, L., 4-wave dynamics in kinetic wave turbulence. Physica D, 24-59 (2018) · Zbl 1386.82051 |
| [166] | Lukkarinen, J.; Spohn, H., Weakly nonlinear Schrödinger equation with random initial data. Invent. Math., 79-188 (2011) · Zbl 1235.76136 |
| [167] | Choi, Y.; Lvov, Y. V.; Nazarenko, S., Joint statistics of amplitudes and phases in wave turbulence. Physica D, 1-2, 121-149 (2005) · Zbl 1143.76451 |
| [168] | Choi, Y.; Lvov, Y. V.; Nazarenko, S., Probability densities and preservation of randomness in wave turbulence. Phys. Lett. A, 230 (2004) · Zbl 1123.76332 |
| [169] | Cercignani, C., On the Boltzmann equation for rigid spheres. Transport Theory Statist. Phys., 3, 211-225 (1972) · Zbl 0295.76048 |
| [170] | Buckmaster, T.; Germain, P.; Hani, Z.; Shatah, J., Onset of the wave turbulence description of the longtime behavior of the nonlinear Schrödinger equation. Invent. Math., 787-855 (2021) · Zbl 1484.35345 |
| [171] | Deng, Y.; Hani, Z., Propagation of chaos and the higher order statistics in the wave kinetic theory (2021), arXiv preprint arXiv:2110.04565 |
| [172] | Dymov, A.; Kuksin, S., Formal expansions in stochastic model for wave turbulence 1: kinetic limit. Comm. Math. Phys., 2, 951-1014 (2021) · Zbl 1482.35211 |
| [173] | Dymov, A.; Kuksin, S.; Maiocchi, A.; Vladuts, S., The large-period limit for equations of discrete turbulence (2021), arXiv preprint arXiv:2104.11967 |
| [174] | Dymov, A.; Kuksin, S., Formal expansions in stochastic model for wave turbulence 2: method of diagram decomposition. J. Stat. Phys., 1, 1-42 (2023) · Zbl 1504.35474 |
| [175] | Peierls, R., Zur kinetischen theorie der wärmeleitung in kristallen. Ann. Phys., 8, 1055-1101 (1929) · JFM 55.0547.01 |
| [176] | Zaslavskii, G.; Sagdeev, R., Limits of statistical description of a nonlinear wave field. Sov. Phys. JETP, 718-724 (1967) · Zbl 0165.29302 |
| [177] | Tanaka, M.; Yokoyama, N., Numerical verification of the random-phase-and-amplitude formalism of weak turbulence. Phys. Rev. E, 6 (2013) |
| [178] | Yokoyama, N.; Takaoka, M., Weak and strong wave turbulence spectra for elastic thin plate. Phys. Rev. Lett., 10 (2013) |
| [179] | Chibbaro, S.; Dematteis, G.; Josserand, C.; Rondoni, L., Wave-turbulence theory of four-wave nonlinear interactions. Phys. Rev. E, 2 (2017) |
| [180] | Zhu, Y.; Semisalov, B.; Krstulovic, G.; Nazarenko, S., Testing wave turbulence theory for the gross-pitaevskii system. Phys. Rev. E, 1 (2022) |
| [181] | Onorato, M.; Dematteis, G.; Proment, D.; Pezzi, A.; Ballarin, M.; Rondoni, L., Negative temperature states as exact equilibrium solutions of the wave kinetic equation for one dimensional lattices (2020), arXiv preprint arXiv:2012.10618 |
| [182] | Balescu, R., Statistical Dynamics: Matter Out of Equilibrium (1997), World Scientific · Zbl 0997.82505 |
| [183] | Kartashova, E., Nonlinear resonance analysis, 2010 |
| [184] | Chirikov, B. V., A universal instability of many-dimensional oscillator systems. Phys. Rep., 5, 263-379 (1979) |
| [185] | L’vov, V. S.; Nazarenko, S., Spectrum of kelvin-wave turbulence in superfluids. JETP Lett., 428-434 (2010) |
| [186] | Laurie, J.; Bortolozzo, U.; Nazarenko, S.; Residori, S., One-dimensional optical wave turbulence: experiment and theory. Phys. Rep., 4, 121-175 (2012) |
| [187] | Fu, W.; Zhang, Y.; Zhao, H., Nonintegrability and thermalization of one-dimensional diatomic lattices. Phys. Rev. E, 5 (2019) |
| [188] | Gorbach, A. V.; Johansson, M., Discrete gap breathers in a diatomic Klein-Gordon chain: Stability and mobility. Phys. Rev. E, 6 (2003) |
| [189] | Koukouloyannis, V.; Kevrekidis, P. G., On the stability of multibreathers in Klein-Gordon chains. Nonlinearity, 9, 2269 (2009) · Zbl 1179.37103 |
| [190] | De Luca, A.; Mussardo, G., Equilibration properties of classical integrable field theories. J. Stat. Mech. Theory Exp., 6 (2016) · Zbl 1456.82243 |
| [191] | Danieli, C.; Mithun, T.; Kati, Y.; Campbell, D. K.; Flach, S., Dynamical glass in weakly nonintegrable Klein-Gordon chains. Phys. Rev. E, 3 (2019) |
| [192] | Parisi, G., On the approach to equilibrium of a Hamiltonian chain of anharmonic oscillators. Europhys. Lett., 4, 357 (1997) |
| [193] | Ponno, A.; Galgani, L.; Guerra, F., Analytical estimate of stochasticity thresholds in Fermi-Pasta-Ulam and \(\varphi 4\) models. Phys. Rev. E, 6, 7081 (2000) |
| [194] | Giorgilli, A.; Paleari, S.; Penati, T., An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit. Ann. Henri Poincaré, 4, 897-959 (2015) · Zbl 1320.82010 |
| [195] | Fucito, F.; Marchesoni, F.; Marinari, E.; Parisi, G.; Peliti, L.; Ruffo, S.; Vulpiani, A., Approach to equilibrium in a chain of nonlinear oscillators. J. Physique, 5, 707-713 (1982) |
| [196] | Lvov, V. S., Wave Turbulence under Parametric Excitation: Applications to Magnets (2012), Springer Science & Business Media |
| [197] | Zakharov, V. E.; Lvov, V. S.; Starobinets, S. S., Spin-wave turbulence beyond the parametric excitation threshold. Phys.-Usp., 6, 896-919 (1975) |
| [198] | J. Bardeen, J. R.S., Microscopic theory of superconductivity. Phys. Rev. (1957) |
| [199] | Miller, P.; Vladimirova, N.; Falkovich, F., Oscillations in a turbulence-condensate system. Phys. Rev. E (2013) |
| [200] | Dyachenko, S.; Pushkarev, A.; Zakharov, V. E., Optical turbulence: weak turbulence, condensates and collapsing filaments in the nonlinear Schrödinger equation. Physica D, 96-160 (1992) · Zbl 0767.35082 |
| [201] | Vladimirova, N.; Derevyanko, S.; Falkovich, F., Phase transitions in wave turbulence. Phys. Rev. E (2012), (R) |
| [202] | Rieder, Z.; Lebowitz, J.; Lieb, E., Properties of a harmonic crystal in a stationary nonequilibrium state. J. Math. Phys., 5, 1073-1078 (1967) |
| [203] | Nakazawa, H., Energy flow in harmonic linear chain. Progr. Theoret. Phys., 1, 236-238 (1968) |
| [204] | Wang, J.-S.; Wang, J.; Zeng, N., Nonequilibrium green’s function approach to mesoscopic thermal transport. Phys. Rev. B, 3 (2006) |
| [205] | Yamamoto, T.; Watanabe, K., Nonequilibrium green’s function approach to phonon transport in defective carbon nanotubes. Phys. Rev. Lett., 25 (2006) |
| [206] | Dhar, A.; Roy, D., Heat transport in harmonic lattices. J. Stat. Phys., 4, 801-820 (2006) · Zbl 1107.82038 |
| [207] | Dhar, A.; Saito, K., Heat transport in harmonic systems, 39-105 |
| [208] | Feynman, R. P.; Leighton, R. B.; Sands, M., The Feynman Lectures on Physics, Vol. I: The New Millennium Edition: Mainly Mechanics, Radiation, and Heat, Vol. 1 (2011), Basic books |
| [209] | baron de Fourier, J. B.J., Théorie Analytique de la Chaleur (1822), Firmin Didot |
| [210] | Iubini, S.; Lepri, S.; Livi, R.; Politi, A.; Politi, P., Nonequilibrium phenomena in nonlinear lattices: From slow relaxation to anomalous transport, 185-203 |
| [211] | Peierls, R., Ann. Phys. (1929) |
| [212] | Ziman, J. M., Electrons and Phonons: The Theory of Transport Phenomena in Solids (2001), Oxford University Press · Zbl 0088.24004 |
| [213] | Ziman, J. M., Principles of the Theory of Solids (1972), Cambridge University Press · Zbl 0121.44801 |
| [214] | Lukkarinen, J.; Spohn, H., Anomalous energy transport in the FPU-\( \beta\) chain. Commun. Pure Appl. Math.: J. Issued Courant Inst. Math. Sci., 12, 1753-1786 (2008) · Zbl 1214.82057 |
| [215] | Herring, C., Role of low-energy phonons in thermal conduction. Phys. Rev., 4, 954 (1954) · Zbl 0058.23706 |
| [216] | Klemens, P. G., Thermal conductivity of solids at low temperatures, 198-281 · Zbl 0043.44103 |
| [217] | Beck, H., Dynamical Properties of Solids (1975), North-Holland, Amsterdam |
| [218] | Nickel, B., The solution to the 4-phonon Boltzmann equation for a 1D chain in a thermal gradient. J. Phys. A, 6, 1219 (2007) · Zbl 1129.82026 |
| [219] | Pomeau, Y.; Resibois, P., Time dependent correlation functions and mode-mode coupling theories. Phys. Rep., 2, 63-139 (1975) |
| [220] | Kubo, R.; Toda, M.; Hashitsume, N., Statistical Physics II: Nonequilibrium Statistical Mechanics, Vol. 31 (2012), Springer Science & Business Media |
| [221] | Ernst, M., Mode-coupling theory and tails in CA fluids. Physica D, 1-2, 198-211 (1991) · Zbl 0718.76003 |
| [222] | Lepri, S., Relaxation of classical many-body Hamiltonians in one dimension. Phys. Rev. E, 6, 7165 (1998) |
| [223] | Delfini, L.; Lepri, S.; Livi, R.; Politi, A., Self-consistent mode-coupling approach to one-dimensional heat transport. Phys. Rev. E, 6 (2006) |
| [224] | Marconi, U. M.B.; Puglisi, A.; Rondoni, L.; Vulpiani, A., Fluctuation-dissipation: response theory in statistical physics. Phys. Rep., 4-6, 111-195 (2008) |
| [225] | Lepri, S.; Livi, R.; Politi, A., Universality of anomalous one-dimensional heat conductivity. Phys. Rev. E, 6 (2003) |
| [226] | Wang, L.; Wang, T., Power-law divergent heat conductivity in one-dimensional momentum-conserving nonlinear lattices. Europhys. Lett., 5, 54002 (2011) |
| [227] | Cipriani, P.; Denisov, S.; Politi, A., From anomalous energy diffusion to levy walks and heat conductivity in one-dimensional systems. Phys. Rev. Lett., 24 (2005) |
| [228] | Zhao, H., Identifying diffusion processes in one-dimensional lattices in thermal equilibrium. Phys. Rev. Lett., 14 (2006) |
| [229] | Zaburdaev, V.; Denisov, S.; Hänggi, P., Perturbation spreading in many-particle systems: A random walk approach. Phys. Rev. Lett., 18 (2011) |
| [230] | Lepri, S.; Politi, A., Density profiles in open superdiffusive systems. Phys. Rev. E, 3 (2011) |
| [231] | Liu, S.; Hänggi, P.; Li, N.; Ren, J.; Li, B., Anomalous heat diffusion. Phys. Rev. Lett., 4 (2014) |
| [232] | Kundu, A.; Bernardin, C.; Saito, K.; Kundu, A.; Dhar, A., Fractional equation description of an open anomalous heat conduction set-up. J. Stat. Mech. Theory Exp., 1 (2019) · Zbl 1539.82297 |
| [233] | Spohn, H., Fluctuating hydrodynamics approach to equilibrium time correlations for anharmonic chains, 107-158 |
| [234] | Mendl, C. B.; Spohn, H., Dynamic correlators of Fermi-Pasta-Ulam chains and nonlinear fluctuating hydrodynamics. Phys. Rev. Lett., 23 (2013) |
| [235] | Das, S. G.; Dhar, A.; Saito, K.; Mendl, C. B.; Spohn, H., Numerical test of hydrodynamic fluctuation theory in the Fermi-Pasta-Ulam chain. Phys. Rev. E, 1 (2014) |
| [236] | Cividini, J.; Kundu, A.; Miron, A.; Mukamel, D., Temperature profile and boundary conditions in an anomalous heat transport model. J. Stat. Mech. Theory Exp., 1 (2017) |
| [237] | Dhar, A.; Kundu, A.; Kundu, A., Anomalous heat transport in one dimensional systems: A description using non-local fractional-type diffusion equation. Front. Phys., 159 (2019) |
| [238] | Crouseilles, N.; Hivert, H.; Lemou, M., Numerical schemes for kinetic equations in the anomalous diffusion limit. Part II: Degenerate collision frequency. SIAM J. Sci. Comput., 4, A2464-A2491 (2016) · Zbl 1515.35032 |
| [239] | Spohn, H., Nonlinear fluctuating hydrodynamics for anharmonic chains. J. Stat. Phys., 5, 1191-1227 (2014) · Zbl 1291.82119 |
| [240] | Jara, M.; Komorowski, T.; Olla, S., Superdiffusion of energy in a chain of harmonic oscillators with noise. Comm. Math. Phys., 2, 407-453 (2015) · Zbl 1329.82116 |
| [241] | Basile, G.; Bernardin, C.; Jara, M.; Komorowski, T.; Olla, S., Thermal conductivity in harmonic lattices with random collisions, 215-237 |
| [242] | Bonetto, F.; Lebowitz, J. L.; Rey-Bellet, L., Fourier’s law: A challenge to theorists, 128-150 · Zbl 1074.82530 |
| [243] | Saint-Raymond, L., Hydrodynamic Limits of the Boltzmann Equation (2009), Springer Science & Business Media · Zbl 1171.82002 |
| [244] | Lepri, S.; Livi, R.; Politi, A., Studies of thermal conductivity in Fermi-Pasta-Ulam-like lattices. Chaos, 1 (2005) |
| [245] | Ampatzoglou, I.; Collot, C.; Germain, P., Derivation of the kinetic wave equation for quadratic dispersive problems in the inhomogeneous setting (2021), arXiv preprint arXiv:2107.11819 |
| [246] | Chester, M., Second sound in solids. Phys. Rev., 5, 2013 (1963) |
| [247] | Chandrasekharaiah, D., Thermoelasticity with second sound: A review (1986) · Zbl 0588.73006 |
| [248] | Prohofsky, E.; Krumhansl, J., Second-sound propagation in dielectric solids. Phys. Rev., 5A, A1403 (1964) |
| [249] | Kuzkin, V. A.; Krivtsov, A. M., Ballistic resonance and thermalization in the Fermi-Pasta-Ulam-Tsingou chain at finite temperature. Phys. Rev. E, 4 (2020) |
| [250] | Fernando, K. M.; Schelling, P. K., Non-local linear-response functions for thermal transport computed with equilibrium molecular-dynamics simulation. J. Appl. Phys., 21 (2020) |
| [251] | Kuzkin, V. A.; Krivtsov, A. M., Unsteady ballistic heat transport: linking lattice dynamics and kinetic theory. Acta Mech., 5, 1983-1996 (2021) · Zbl 1504.74004 |
| [252] | Bohm, N.; Schelling, P. K., Analysis of ballistic transport and resonance in the \(\alpha \)-Fermi-Pasta-Ulam-Tsingou model. Phys. Rev. E, 2 (2022) |
| [253] | Zaslavskii, G.; Sagdeev, R., Sov. Phys.—JETP, 718 (1967) |
| [254] | Choi, Y.; Lvov, Y. V.; Nazarenko, S.; Pokorni, B., Anomalous probability of large amplitudes in wave turbulence. Phys. Lett. A, 3-5, 361-369 (2005) · Zbl 1145.76394 |
| [255] | Rosenzweig, M.; Staffilani, G., Uniqueness of solutions to the spectral hierarchy in kinetic wave turbulence theory. Physica D (2022) · Zbl 1539.35244 |
| [256] | Guioth, J.; Bouchet, F.; Eyink, G. L., Path large deviations for the kinetic theory of weak turbulence. J. Stat. Phys., 2, 20 (2022) · Zbl 07591719 |
| [257] | Hrabski, A.; Pan, Y., On the properties of energy flux in wave turbulence. J. Fluid Mech. (2022) · Zbl 07477160 |
| [258] | Lieb, E. H.; Mattis, D. C., Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles (2013), Academic Press |
| [259] | Chaikin, P. M.; Lubensky, T. C.; Witten, T. A., Principles of Condensed Matter Physics, Vol. 10 (1995), Cambridge university press Cambridge |
| [260] | Anderson, P. W., Random-phase approximation in the theory of superconductivity. Phys. Rev., 6, 1900 (1958) |
| [261] | Ishii, K., Localization of eigenstates and transport phenomena in the one-dimensional disordered system. Progr. Theoret. Phys. Suppl., 77-138 (1973) |
| [262] | Fröhlich, J.; Spencer, T., Absence of diffusion in the Anderson tight binding model for large disorder or low energy. Comm. Math. Phys., 2, 151-184 (1983) · Zbl 0519.60066 |
| [263] | Benettin, G.; Fröhlich, J.; Giorgilli, A., A Nekhoroshev-type theorem for Hamiltonian systems with infinitely many degrees of freedom. Comm. Math. Phys., 95-108 (1988) · Zbl 0825.58011 |
| [264] | Campbell, D. K.; Flach, S.; Kivshar, Y. S., Localizing energy through nonlinearity and discreteness. Phys. Today, 1, 43-49 (2004) |
| [265] | Flach, S.; Krimer, D.; Skokos, C., Universal spreading of wave packets in disordered nonlinear systems. Phys. Rev. Lett., 2 (2009) |
| [266] | Ivanchenko, M.; Laptyeva, T.; Flach, S., Anderson localization or nonlinear waves: A matter of probability. Phys. Rev. Lett., 24 (2011) |
| [267] | Mulansky, M.; Ahnert, K.; Pikovsky, A.; Shepelyansky, D., Strong and weak chaos in weakly nonintegrable many-body Hamiltonian systems. J. Stat. Phys., 1256-1274 (2011) · Zbl 1252.82033 |
| [268] | Zhang, Z.; Tang, C.; Kang, J.; Tong, P., Dynamical energy equipartition of the Toda model with additional on-site potentials. Chin. Phys. B, 10 (2017) |
| [269] | Sun, L.; Zhang, Z.; Tong, P., Effects of weak disorder on the thermalization of Fermi-Pasta-Ulam-Tsingou model. New J. Phys., 7 (2020) |
| [270] | Wang, Z.; Fu, W.; Zhang, Y.; Zhao, H., Wave-turbulence origin of the instability of anderson localization against many-body interactions. Phys. Rev. Lett., 18 (2020) |
| [271] | Segev, M.; Silberberg, Y.; Christodoulides, D. N., Anderson localization of light. Nature Photonics, 3, 197-204 (2013) |
| [272] | Benettin, G., Time scale for energy equipartition in a two-dimensional FPU model. Chaos, 1 (2005) · Zbl 1080.82009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
