{\rm const}$, the inter-particle noise conserves locally the kinetic energy and momentum.

Concerning the Hamiltonian part of the dynamics, we assume (cf [@BOS]) that the coupling constants $({\alpha}_x)_{x\in{{\mathbb Z}}}$ satisfy the following:

-   they are real valued and there exists $C>0$ such that $|\alpha_x|\le Ce^{-|x|/C}$ for all $x\in {{\mathbb Z}}$,

-   $\hat\alpha(k)$ is also real valued and $\hat\alpha(k)>0$ for $k\not=0$ and in case $\hat \alpha(0)=0$ we have $\hat\alpha''(0)>0$.

The above conditions imply that both functions $x\mapsto\alpha_x$ and $k\mapsto\hat\alpha(k)$ are even. In addition, $\hat\alpha\in
   C^{\infty}({{\mathbb T}})$ and in case $\hat\alpha(0)=0$ we have $\hat\alpha(k)=k^2\phi(k^2)$ for some strictly positive $\phi\in
   C^{\infty}({{\mathbb T}})$. The dispersion relation ${\omega}:{{\mathbb T}}\to
   \bar{{\mathbb R}}_+$, given by $$\label{mar2602}
 {\omega}(k):=\sqrt{\hat \alpha (k)}$$ is even. Throughout the paper it is assumed to be unimodal, i.e. increasing on $[0,1/2]$ and then, in consequence, decreasing on $[-1/2,0]$. Its unique minimum and maximum, attained at $k=0$, $k=1/2$, respectively are denoted by ${\omega}_{\rm min}\ge 0$ and ${\omega}_{\rm max}$, correspondingly. We denote the two branches of its inverse by ${\omega}_\pm:[{\omega}_{\rm min},{\omega}_{\rm max}]\to\bar {{\mathbb T}}_\pm$.

### Initial data

We assume that the initial data is random and, given ${\epsilon}>0$, distributed according to probabilistic measure $\mu_{\epsilon}$ and $$\label{mu-eps}
{\cal E}_*:=\sup_{{\epsilon}\in(0,1]}{\epsilon}\sum_{x\in{{\mathbb Z}}}\langle{\frak e}_x\rangle_{\mu_{\epsilon}}<+\infty.$$ Here $\langle\cdot\rangle_{\mu_{\epsilon}}$ is the expectation with 