Geodesic
On the thermalization of one-dimensional lattices. I.~Microcanonical ensemble
Matematičeskaâ biologiâ i bioinformatika, Tome 19 (2024) no. 1, pp. 248-260

Voir la notice de l'article provenant de la source Math-Net.Ru

In numerical simulation of biomacromolecule, the issues of thermalization, i.e., equal distribution of energy over the degrees of freedom, occupy an important place. In this paper we consider some mechanisms of lattice thermalization: Chirikov resonances, wave turbulence and some others. We consider thermalization in a microcanonical ensemble when the system is isolated from external fields and the total energy is conserved. Although microcanonical ensembles are rarely used in practical calculations, however, the basic ideas about the thermalization mechanisms are obtained for these systems. The main attention is paid to the consideration of the lattices thermalization with Fermi–Pasta–Ulam–Tsingou potentials, since the main efforts to understand the basis of thermalization have been made precisely for lattices of this type. The role of solitons and breathers in thermalization is discussed. 
@article{MBB_2024_19_1_a13,
     author = {G. A. Vinogradov and V. D. Lakhno},
     title = {On the thermalization of one-dimensional lattices. {I.~Microcanonical} ensemble},
     journal = {Matemati\v{c}eska\^a biologi\^a i bioinformatika},
     pages = {248--260},
     publisher = {mathdoc},
     volume = {19},
     number = {1},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MBB_2024_19_1_a13/}
}
TY  - JOUR
AU  - G. A. Vinogradov
AU  - V. D. Lakhno
TI  - On the thermalization of one-dimensional lattices. I.~Microcanonical ensemble
JO  - Matematičeskaâ biologiâ i bioinformatika
PY  - 2024
SP  - 248
EP  - 260
VL  - 19
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MBB_2024_19_1_a13/
LA  - ru
ID  - MBB_2024_19_1_a13
ER  - 
%0 Journal Article
%A G. A. Vinogradov
%A V. D. Lakhno
%T On the thermalization of one-dimensional lattices. I.~Microcanonical ensemble
%J Matematičeskaâ biologiâ i bioinformatika
%D 2024
%P 248-260
%V 19
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MBB_2024_19_1_a13/
%G ru
%F MBB_2024_19_1_a13
G. A. Vinogradov; V. D. Lakhno. On the thermalization of one-dimensional lattices. I.~Microcanonical ensemble. Matematičeskaâ biologiâ i bioinformatika, Tome 19 (2024) no. 1, pp. 248-260. http://geodesic.mathdoc.fr/item/MBB_2024_19_1_a13/

[1] A. S. Shigaev, O. A. Ponomarev, V. D. Lakhno, “Theoretical and experimental investigation of DNA open state”, Mathematical Biology and Bioinformatics, 13:S (2018) | DOI | DOI

[2] I. V. Likhachev, A. S. Shigaev, V. D. Lakhno, “On the thermodynamics of DNA double strand in the Peyrard-Bishop-Dauxois model”, Phys. Lett. A, 510 (2024), 129547 | DOI | MR | DOI | MR

[3] A. Ya. Khinchin, Matematicheskie osnovaniya statisticheskoi mekhaniki, Nauka, M., 2003 | MR | MR

[4] R. Kubo, M. Toda, N. Hashitsume, Statistical Physics II: Nonequilibrium Statistical Mechanics, Springer Series in Solid-State Science, 31, Springer, NewYork, 1985 | DOI | MR | DOI | MR

[5] M. Toda, Theory of Nonlinear Lattices, Springer, New York, 1989 | MR | Zbl | MR | Zbl

[6] V. N. Likhachev, T. Yu. Astakhova, G. A. Vinogradov, “Thermodynamics and ergodicity of finite one-dimensional Toda and Morse lattices”, Phys. Lett. A, 354 (2006), 264–270 | DOI | MR | Zbl | DOI | MR | Zbl

[7] E. Fermi, “Dimostrazione che in generale un sistema meccanico normale e quasi ergodico”, Il Nuovo Cimento, 25 (1923), 267–269 (In Italian) | DOI | DOI

[8] E. Fermi, J. Pasta, S. Ulam, Los Alamos Report LA No 1940, 1955 | Zbl | Zbl

[9] The Collected Papers of Enrico Fermi, v. II, Chicago University Press, Chicago, 1965 | MR | MR

[10] D. K. Campbell, P. Rosenau, G. M. Zaslavsky, “Introduction: The Fermi-Pasta-Ulam problem-The first fifty years”, Chaos, 15:1 (2005) | DOI | Zbl | DOI | Zbl

[11] Gallavotti G. (ed.), The Fermi-Pasta-Ulam Problem: A Status Report, Lecture Notes in Physics, 728, Springer, 2008, 308 pp. | DOI | MR | Zbl | DOI | MR | Zbl

[12] T. Dauxois, “Fermi, Pasta, Ulam, a mysterious lady”, Physics Today, 61:1 (2008), 55–57 | DOI | DOI

[13] V. E. Zakharov, E. I. Schulman, “Integrability of nonlinear systems and perturbation theory”, What is integrability?, ed. Zakharov V.E., Springer, Berlin, 1991, 185–250 | DOI | MR | Zbl | DOI | MR | Zbl

[14] N. J. Zabusky, M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and the reccurence of initial states”, Phys. Rev. Lett, 15:6 (1965), 240–243 | DOI | Zbl | DOI | Zbl

[15] S. D. Pace, K. A. Reiss, D. K. Campbell, “The-Fermi-Pasta-Ulam-Tsingou recurrence problem”, Chaos, 29:11 (2019), 113107 | DOI | MR | Zbl | DOI | MR | Zbl

[16] D. Pierangeli, M. Flammini, L. Zhang, Marcucci G. Agranat, A. J., P. G. Grinevich, P. M. Santini, C. Conti, E. DelRe, “Observation of Fermi-Pasta-Ulam-Tsingou Recurrence and Its Exact Dynamics”, Phys. Rev. X, 8 (2018), 041017 | DOI | DOI

[17] A. Hofstrand, “Near-integrable dynamics of the Fermi-Pasta-Ulam-Tsingou problem”, Phys. Rev. E, 109 (2024), 034204 | DOI | MR | DOI | MR

[18] F. Borgonovi, F. M. Izrailev, “Classical statistical mechanics of a few-body interacting spin model”, Phys. Rev. E, 62 (2000), 6475 | DOI | MR | DOI | MR

[19] F. M. Izrailev, B. V. Chirikov, Dokl. AN SSSR, 166 (1966), 57 | Zbl | Zbl

[20] B. V. Chirikov, Phys. Rep, 52 (1979), 263 | DOI | MR | DOI | MR

[21] V. N. Likhachev, T. Yu. Astakhova, G. A. Vinogradov, “The Dynamics of a One-Dimensional Disordered Morse Lattice: Cooling Caused by Adiabatic Stretching”, Russ. J. Phys. Chem. B, 3:1 (2009), 140–150 | DOI | MR | DOI | MR

[22] M. Onorato, Y. V. Lvov, “Double Scaling in the Relaxation Time in the-Fermi-Pasta-Ulam-Tsingou Model”, Phys. Rev. Lett, 120 (2018), 144301 | DOI | DOI

[23] M. Onorato, L. Vozella, D. Proment, Y. V. Lvov, “Route to thermalization in the-Fermi-Pasta-Ulam system”, PNAS, 112:14 (2015), 4208–4213 | DOI | DOI

[24] G. Benettin, H. Christodoulidi, A. Ponno, “The Fermi-Pasta-Ulam problem and its underlying integrable dynamics”, J. Stat. Phys, 152:2 (2013), 195–212 | DOI | MR | DOI | MR

[25] V. V. Kozlov, Teplovoe ravnovesie po Gibbsu i Puankare, Institut kompyuternykh issledovanii, Moskva-Izhevsk, 2002 | MR | MR

[26] V. E. Zakharov, Y. S. L'vov, G. Falkovich, Kolmogorov spectra of turbulence I: Wave turbulence, Springer Science Business Media, 2012 | MR | MR

[27] Galtier S., Physics of Wave Turbulence, Cambridge University Press, 2022

[28] M. Onorato, Y. V. Lvov, G. Dematteis, S. Chibbaro, “Wave Turbulence and thermalization in one-dimensional chains”, Phys. Reports, 1040 (2023), 1–36 | DOI | MR | Zbl | DOI | MR | Zbl

[29] Y. Liu, D. He, “Analytical approach to Lyapunov time: Universal scaling and thermalization”, Phys. Rev. E, 103 (2021) | DOI | MR | DOI | MR

[30] M. Pettini, Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics, Springer, NewYork, 2008 | MR | MR

[31] N. S. Krylov, Works on the Foundations of Statistical Physics, Princeton University Press, Princeton, 1979 | MR | MR

[32] A. S. Davydov, Solitony v molekulyarnykh sistemakh, Nauk. dumka, Kiev, 1984 | MR | MR

[33] P. S. Lomdahl, W. C. Kerr, Phys. Rev. Lett., 55 (1985), 1235 | DOI | DOI

[34] X. D. Wang, D. W. Brown, K. Lindenberg, Phys. Rev. Lett, 62 (1989), 1796 | DOI | DOI

[35] S. Flach, C. R. Willis, “Discrete breathers”, Phys. Reports, 295 (1998), 181–264 | DOI | MR | DOI | MR

[36] S. Aubry, “Discrete Breathers: Localization and transfer of energy in discrete Hamiltonian nonlinear systems”, Physica D, 216 (2006), 1–30 | DOI | MR | Zbl | DOI | MR | Zbl

[37] S. Flach, A. V. Gorbach, “Discrete breathers- Advances in theory and applications”, Phys. Reports, 467 (2008), 1–116 | DOI | DOI

[38] R. S. MacKay, S. Aubry, “Discrete breathers in anharmonic models with acoustic phonons”, Nonlinearity, 7 (1994), 1623–1643 | DOI | MR | Zbl | DOI | MR | Zbl

[39] A. S. Davydov, Solitony v bioenergetike, Nauk. dumka, Kiev, 1986 | MR | MR

[40] G. Kopidakis, S. Aubry, G. P. Tsironis, “Targeted Energy Transfer through Discrete Breathers in Nonlinear Systems”, Phys. Rev. Lett, 87:16 (2001), 165501 | DOI | MR | DOI | MR

[41] T. Yu. Astakhova, V. N. Likhachev, G. A. Vinogradov, Heat conductance in nonlinear lattices at small temperature gradients, arXiv: 1006.1779

[42] S. Lepri, R. Livi, A. Politi, “Thermal conduction in classical low-dimensional lattices”, Phys. Reports, 377 (2003), 1–80 | DOI | MR | DOI | MR