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Nonequilibrium statistical mechanics of Hamiltonian rotators with alternated spins. (English) Zbl 1328.82046
The author considers a Hamiltonian system consisting of a \(d\)-dimensional lattice of \(N\) nonlinear rotators, with neighboring rotators having opposite spin. The rotators are weakly coupled through a potential (linear or nonlinear) of size \(\varepsilon^a\), with \(a\geq 1/2\). Each rotator of the Hamiltonian system interacts with its own stochastic Langevin-type thermostat with a force of order \(\varepsilon\). Introducing action-angle variables for the uncoupled Hamiltonian system (corresponding to \(\varepsilon=0\)), it can be noted that the sum of the actions is conserved by the Hamiltonian dynamics, therefore the actions play the role of the local energy. The author investigates the limiting dynamics (for \(\varepsilon \to 0\)) of the actions of the \(\varepsilon\)-perturbed system on time intervals of order \(\epsilon^{-1}\), showing that it is governed by an autonomous system of stochastic equations, which has a completely non-Hamiltonian character. The Hamiltonian system coupled with the thermal baths is mixing and the author shows that its stationary measure converges to the product of the unique stationary measure of the autonomous system and the normalized Lebesgue measure on the \(N\)-dimensional torus. It is also proved that as \(\varepsilon \to 0\) the convergence of the vector of the actions to a solution of the autonomous system is uniform with respect to \(N\), while the convergence of the measures is uniform only in some natural cases.

82C70 Transport processes in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
34E10 Perturbations, asymptotics of solutions to ordinary differential equations
Full Text: DOI
[1] Basile, G; Bernardin, C; Olla, S, Thermal conductivity for a momentum conservative model, Commun. Math. Phys., 287, 67-98, (2009) · Zbl 1178.82070
[2] Basile, G; Olla, S; Spohn, H, Energy transport in stochastically perturbed lattice dynamics, Arch. Rat. Mech. Anal., 195, 171-203, (2010) · Zbl 1187.82017
[3] Bernardin, C; Huveneers, F, Small perturbation of a disordered harmonic chain by a noise and an anharmonic potential, Probab. Theory Relat. Fields, 157, 301-331, (2013) · Zbl 1281.82026
[4] Bernardin, C; Olla, S, Fourier’s law for a microscopic model of heat conduction, J. Statist. Phys., 118, 271-289, (2005) · Zbl 1127.82042
[5] Bernardin, C., Huveneers, F., Lebowitz, J.L., Liverani, C., Olla, S.: Green-Kubo formula for weakly coupled system with dynamical noise (2013). arXiv:1311.7384v1 · Zbl 1311.82038
[6] Bernardin, C; Kannan, V; Lebowitz, JL; Lukkarinen, J, Harmonic systems with bulk noises, J. Statist. Phys., 146, 800-831, (2011) · Zbl 1242.82034
[7] Bonetto, F; Lebowitz, JL; Lukkarinen, J, Fourier’s law for a harmonic crystal with self-consistent stochastic reservoirs, J. Statist. Phys., 116, 783-813, (2004) · Zbl 1142.82367
[8] Bonetto, F., Lebowitz, J.L., Rey-Bellet, L.: Fourier’s law: a challenge to theorists. In: Mathematical Physics, pp. 128-150. Imperial College Press, London (2000) · Zbl 1074.82530
[9] Bonetto, F; Lebowitz, JL; Lukkarinen, J; Olla, S, Heat conduction and entropy production in anharmonic crystals with self-consistent stochastic reservoirs, J. Statist. Phys., 134, 1097-1119, (2009) · Zbl 1173.82017
[10] Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996) · Zbl 0849.60052
[11] Dolgopyat, D; Liverani, C, Energy transfer in a fast-slow Hamiltonian system, Commun. Math. Phys., 308, 201-225, (2011) · Zbl 1235.82065
[12] Dudley, R.M.: Real Analysis and Probability. Cambridge University Press, Cambridge (2002) · Zbl 1023.60001
[13] Dymov, AV, Dissipative effects in a linear Lagrangian system with infinitely many degrees of freedom, Izv. Math., 76, 1116-1149, (2012) · Zbl 1261.37032
[14] Dymov, A.: Statistical mechanics of nonequilibrium systems of rotators with alternated spins (2014). arXiv:1403.1219 · Zbl 1328.82046
[15] Eckmann, J-P; Pillet, C-A; Rey-Bellet, L, Non-equilibrium statistical mechanics of anharmonic chains coupled to two heat baths at different temperatures, Commun. Math. Phys., 201, 657-697, (1999) · Zbl 0932.60103
[16] Freidlin, M; Wentzell, A, Averaging principle for stochastic perturbations of multifrequency systems, Stoch. Dyn., 3, 393-408, (2003) · Zbl 1050.60078
[17] Freidlin, MI; Wentzell, AD, Long-time behavior of weakly coupled oscillators, J. Statist. Phys., 123, 1311-1337, (2006) · Zbl 1119.34044
[18] Freidlin, M., Wentzell, A.: Random Perturbations of Dynamical Systems, 3rd edn. Springer, Berlin (2012) · Zbl 1267.60004
[19] Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer, Berlin (1991) · Zbl 0734.60060
[20] Khasminskii, RZ, On the averaging principle for ito stochastic differential equations, Kybernetika, 4, 260-279, (1968) · Zbl 0231.60045
[21] Khasminskii, R.: Stochastic Stability of Differential Equations, 2nd edn. Springer, Berlin (2012) · Zbl 1241.60002
[22] Krylov, N.V.: Controlled Diffusion Processes. Springer, Berlin (1980) · Zbl 0459.93002
[23] Kuksin, SB, Damped-driven KdV and effective equations for long-time behaviour of its solutions, GAFA, 20, 1431-1463, (2010) · Zbl 1231.35205
[24] Kuksin, SB, Weakly nonlinear stochastic CGL equations, Ann. IHP PR, 49, 1033-1056, (2013) · Zbl 1280.35144
[25] Kuksin, SB; Piatnitski, AL, Khasminskii-witham averaging for randomly perturbed KdV equation, J. Math. Pures Appl., 89, 400-428, (2008) · Zbl 1148.35077
[26] Kuksin, S., Shirikyan, A.: Mathematics of Two-Dimensional Turbulence. Cambridge University Press, Cambridge (2012) · Zbl 1333.76003
[27] Liverani, C; Olla, S, Toward the Fourier law for a weakly interacting anharmonic crystal, AMS, 25, 555-583, (2012) · Zbl 1245.82062
[28] Moser, J., Siegel, C.L.: Lectures on Celestial Mechanics. Springer, Berlin (1971) · Zbl 0312.70017
[29] Rey-Bellet, L; Thomas, LE, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics, Commun. Math. Phys., 225, 305-329, (2002) · Zbl 0989.82023
[30] Rockner, M; Schmuland, B; Zhang, X, Yamada-Watanabe theorem for stochastic evolution equations in infinite dimensions, Condens. Matter Phys., 11, 247-259, (2008)
[31] Ruelle, DA, Mechanical model for fourier’s law of heat conduction, Commun. Math. Phys., 311, 755-768, (2012) · Zbl 1246.82075
[32] Shirikyan, A, Local times for solutions of the complex Ginzburg-Landau equation and the inviscid limit, J. Math. Anal. Appl., 384, 130-137, (2011) · Zbl 1248.60075
[33] Temirgaliev, N, A connection between inclusion theorems and the uniform convergence of multiple Fourier series, Math. Notes Acad. Sci. USSR, 12, 518-523, (1972) · Zbl 0253.42022
[34] Treschev, D, Oscillator and thermostat, Discrete Contin. Dyn. Syst., 28, 1693-1712, (2010) · Zbl 1209.70014
[35] Veretennikov, A, Bounds for the mixing rate in the theory of stochastic equations, Theory Probab. Appl., 32, 273-281, (1987) · Zbl 0663.60046
[36] Veretennikov, AY, On polynomial mixing bounds for stochastic differential equations, Stoch. Proc. Their Appl., 70, 115-127, (1997) · Zbl 0911.60042
[37] Whitney, H, Differentiable even functions, Duke Math. J., 10, 159-160, (1942) · Zbl 0063.08235
[38] Yamada, T; Watanabe, S, On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ., 11, 155-167, (1971) · Zbl 0236.60037
[39] Yor, M, Existence et unicité de diffusion à valeurs dans un espace de Hilbert, Ann. Inst. Henri Poincaré Sec. B, 10, 55-88, (1974) · Zbl 0281.60094
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