Nonequilibrium statistical mechanics of weakly stochastically perturbed system of oscillators

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.