Abstract
We consider a finite region of a d-dimensional lattice, \({d \in \mathbb{N}}\), of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size \({\varepsilon}\). Each oscillator weakly interacts by force of order \({\varepsilon}\) with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as \({\varepsilon \rightarrow 0}\) behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order \({\varepsilon^{-1}}\) and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next, we assume that the interaction potential is of size \({\varepsilon \lambda}\), where \({\lambda}\) is another small parameter, independent from \({\varepsilon}\). Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit \({\varepsilon \rightarrow 0}\), the main order in \({\lambda}\) of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space–time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system.
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Communicated by Christian Maes.
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Dymov, A. Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed System of Oscillators. Ann. Henri Poincaré 17, 1825–1882 (2016). https://doi.org/10.1007/s00023-015-0441-x
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DOI: https://doi.org/10.1007/s00023-015-0441-x