Nonequilibrium Statistical Mechanics of Weakly Stochastically Perturbed System of Oscillators

We consider a finite region of a d-dimensional lattice, , of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size . Each oscillator weakly interacts by force of order with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order 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 , where is another small parameter, independent from . Solutions corresponding to this scaling describe small low temperature oscillations. We prove that in a stationary regime, under the limit , the main order in 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.