that the total kinetic energy and momentum are conserved]{}. This stochastic exchange can be modeled in various ways: e.g. for each couple of nearest neighbor particles the exchange of their momenta can occur independently at an exponential time (which models their elastic collision). Otherwise, for each triple of consecutive particles, exchange of momenta can be performed in a continuous, diffusive fashion, so that its energy and momentum are preserved. In the present article we adopt the latter choice, see Section \[sec2.2.1\] for a detailed description of the dynamics, but, with no significant changes, all our results can be extended to other stochastic noises. A small parameter ${\epsilon}>0$ is introduced to rescale space and time, and the intensity of the noise is adjusted so that in a (macroscopic) finite interval of time, there is only a finite amount of momentum exchanged by the stochastic mechanism. In terms of the random exchanges, it means that, on average, each particle undergoes only a finite number of stochastic collisions in a finite time. Letting ${\epsilon}\to 0$ corresponds therefore to taking the kinetic limit for the system.

The Wigner distribution is a useful tool to localize in space the energy per frequency mode. In the 