of the bulk noise.

The goal of the present paper is to extend the result of [@kors] to the case when the inter-particle noise is present, i.e. ${\gamma}_0>0$, see Theorem \[main-thm\] below for the precise formulation of our main result. We emphasize that in the situation when $\gamma_0=0$, both the equations for the microscopic and macroscopic dynamics, given below by and respectively, can be solved explicitly, in terms of the initial condition, and this fact has been extensively used in the proof in [@kors]. The argument can be extended to the dynamics where only the damping terms of the noise are present, i.e. with no noise input both from the inter-particle scattering and the thermostat, see . The equation for the macroscopic limit of the respective Wigner distribution $W^{\rm un}(t,y,k)$ reads (cf ), see Theorem \[main:thm-un\] below, $$\label{eq:damp}
   \partial_t W^{\rm un}(t,y,k) + \bar\omega'(k) \partial_y W^{\rm un}(t,y,k) =
  -2 \gamma_0 R(k) W^{\rm un}(t,y,k), \quad y\not=0,$$ with the boundary conditions as in . In the next step we add the stochastic part corresponding to the inter-particle scattering, which corresponds to $T=0$ for the thermostat, see equation formulated for the respective wave function. Next, we use the previously described dynamics to represent the 