Geodesic
Large system of oscillators with ultralocal stochastic stationary external field influence
Čebyševskij sbornik, Tome 23 (2022) no. 1, pp. 130-141.

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

In this paper influence of external force, assumed to be random stationary process, on the behavior of large Hamiltonian particle systems is studied. The Hamiltonian system is assumed to have quadratic interaction, and the external influence is assumed to be local. More exactly, the external force acts on only one fixed particle. Such systems were studed earlier, it is given short review of the previous papers. In our case, when the external force is a stationary random process in the wider sense, large time asymptotics of the mean energy of the system is studied. Main result is the characterization of 4 different cases for the spectrum of the matrix of quadratic interaction and the spectral density of the correlation function of the stationary random process, which give different asymptotic behaviour of the trajectories and of the mean energy. Typical behaviour appears to be either uniform boundedness or quadratic growth of the mean energies.
Keywords: stationary random processes, linear Hamiltonian systems, local external influence, resonance, asymptotics of mean energies. 
@article{CHEB_2022_23_1_a9,
     author = {M. V. Melikyan},
     title = {Large system of oscillators with ultralocal stochastic stationary external field influence},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {130--141},
     publisher = {mathdoc},
     volume = {23},
     number = {1},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2022_23_1_a9/}
}
TY  - JOUR
AU  - M. V. Melikyan
TI  - Large system of oscillators with ultralocal stochastic stationary external field influence
JO  - Čebyševskij sbornik
PY  - 2022
SP  - 130
EP  - 141
VL  - 23
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2022_23_1_a9/
LA  - ru
ID  - CHEB_2022_23_1_a9
ER  - 
%0 Journal Article
%A M. V. Melikyan
%T Large system of oscillators with ultralocal stochastic stationary external field influence
%J Čebyševskij sbornik
%D 2022
%P 130-141
%V 23
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2022_23_1_a9/
%G ru
%F CHEB_2022_23_1_a9
M. V. Melikyan. Large system of oscillators with ultralocal stochastic stationary external field influence. Čebyševskij sbornik, Tome 23 (2022) no. 1, pp. 130-141. http://geodesic.mathdoc.fr/item/CHEB_2022_23_1_a9/

[1] Lykov A., Malyshev V., “From the $N$-body problem to Euler equations”, Russian Journal of Mathematical Physics, 24:1 (2017), 79–95 | DOI | MR | Zbl

[2] Lykov A. A., Malyshev V. A., “Harmonic Chain with Weak Dissipation”, Markov Processes and Related Fields, 18 (2012), 1–10 | MR

[3] Lykov A. A., Malyshev V. A., Chubarikov V. N., “Regulyarnye kontinualnye sistemy tochechnykh chastits. I: sistemy bez vzaimodeistviya”, Chebyshevskii sbornik, 17:3 (2016), 148–165 | DOI | MR | Zbl

[4] Lykov A. A., Malyshev V. A., “Convergence to Gibbs Equilibrium — Unveiling the Mystery”, Markov Processes and Related Fields, 19 (2013), 643–666 | MR | Zbl

[5] Lykov A. A., Malyshev V. A., Melikian M. V., “Phase diagram for one-way traffic flow with local control”, Physica A: Statistical Mechanics and its Applications, 486 (2017), 849–866 | DOI | MR | Zbl

[6] Lykov A., Melikian M., “Long time behavior of infinite harmonic chain with $l_{2}$ initial conditions”, Markov Processes and Related Fields, 26:2 (2020), 189–212 | MR | Zbl

[7] Lykov A. A., Malyshev V. A., Melikyan M. V., “Rezonans v mnogokomponentnykh lineinykh sistemakh”, Vestnik Moskovskogo universiteta. Seriya 1. Matematika. Mekhanika, 2021, no. 3, 74–79 | MR | Zbl

[8] Dobrushin R. L., Fritz J., “Non-Equilibrium Dynamics of One-dimensional Infinite Particle Systems with a Hard-Core Interaction”, Commun. math. Phys., 55 (1977), 275–292 | DOI | MR | Zbl

[9] Boldrighini C., Pellegrinotti A., Triolo L., “Convergence to Stationary States for Infinite Harmonic Systems”, Journal of Statistical Physics, 30:1 (1983) | DOI | MR

[10] Boldrighini C., Dobrushin R. L., Sukhov Yu. M., “One-Dimensional Hard Rod Caricature of Hydrodynamics”, Journal of Statistical Physics, 31:3 (1983), 123–155 | DOI | MR

[11] Dobrushin R. L. , Pellegrinotti A., Suhov Yu. M., Triolo L., “One-Dimensional Harmonic Lattice Caricature of Hydrodynamics”, Journal of Statistical Physics, 43:3/4 (1986) | MR | Zbl

[12] Bernardin C., Huveneers F. , Olla S., “Hydrodynamic Limit for a Disordered Harmonic Chain”, Commun. Math. Phys., 365 (2019), 215 | DOI | MR | Zbl

[13] Dyson F. J., “The dynamics of a disordered linear chain”, Phys. Rev., 92:6 (1953), 1331–1338 | DOI | MR | Zbl

[14] Matsuda H., Ishii K., “Localization of normal modes and energy transport in the disordered harmonic chain”, Prog. Theor. Phys. Suppl., 45 (1970), 56–88 | DOI | MR

[15] O'Connor A. J., Lebowitz J. L., “Heat conduction and sound transmission in isotopically disordered harmonic crystals”, J. Math. Phys., 15 (1974), 692–703 | DOI | MR

[16] Casher A., Lebowitz J. L., “Heat flow in regular and disordered harmonic chains”, J. Math. Phys., 12:8 (1971), 1701–1711 | DOI

[17] Dudnikova T. V., “Behavior for Large Time of a Two-Component Chain of Harmonic Oscillators”, Russian Journal of Mathematical Physics, 25:4 (2018), 470–491 | DOI | MR | Zbl

[18] Dudnikova T., “Long-time asymptotics of solutions to a hamiltonian system on a lattice”, Journal of Mathematical Sciences, 219:1 (2016), 69–85 | DOI | MR | Zbl

[19] Dudnikova T., Komech A., Spohn H., “On the convergence to statistical equilibrium for harmonic crystals”, J. Math. Phys., 44:6 (2003), 2596–2620 | DOI | MR | Zbl

[20] Lykov A. A., “Energy Growth of Infinite Harmonic Chain under Microscopic Random Influence”, Markov Processes and Related Fields, 26 (2020), 287–304 | MR | Zbl

[21] Kuzkin V. A., Krivtsov A. M., “Energy transfer to a harmonic chain under kinematic and force loadings: Exact and asymptotic solutions”, J. Micromech. and Mol. Phys., 3 (2018), 1–2 | DOI

[22] Hemmen J., “Dynamics and ergodicity of the infinite harmonic crystal”, Physics Reports, 65:2 (1980), 43–149 | DOI | MR

[23] Fox R., “Long-time tails and diffusion”, Phys. Rev. A, 27:6 (1983), 3216–3233 | DOI | MR

[24] Florencio J., Lee H., “Exact time evolution of a classical harmonic-oscillator chain”, Phys. Rev. A, 31:5 (1985), 3221–3236 | DOI | MR

[25] Lanford O., Lebowitz J., “Time evolution and ergodic properties of harmonic systems”, Dynamical Systems, Theory and Applications, Lect. Notes Phys., 38, ed. J. Moser, Springer, Berlin–Heidelberg, 1975, 144–177 | DOI | MR

[26] Bogolyubov N. N., On Some Statistical Methods in Mathematical Physics, Ac. Sci. USSR, Kiev, 1945 | MR

[27] Spohn H., Lebowitz J., “Stationary non-equilibrium states of infinite harmonic systems”, Commun. Math. Phys., 54 (1977), 97–120 | DOI | MR

[28] Filippov A. F., Vvedenie v teoriyu differentsialnykh uravnenii, KomKniga, M., 2007