check that $$\label{e1}
\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}=\|\psi^{({\epsilon})}(t)\|^2_{\ell^2}=\sum_{x\in{{\mathbb Z}}}\left({\frak p}^{({\epsilon})}_x(t)\right)^2+\sum_{x,x'\in{{\mathbb Z}}}{\alpha}_{x-x'}{\frak q}^{({\epsilon})}_x(t){\frak q}^{({\epsilon})}_{x'}(t).$$ After straightforward calculations one can verify that $$\label{031709}
 d\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}
 =-\frac{2{\gamma}_1}{{\epsilon}} \left[
 \left({\frak p}^{({\epsilon})}_0(t)\right)^2
 - T \right]dt
  +\left(\frac{2{\gamma}_1T}{{\epsilon}}\right)^{1/2}{\frak p}^{({\epsilon})}_0(t) dw(t),\quad {{\mathbb P}}_{\epsilon}\,\mbox{-a.s.}$$ Here ${{\mathbb P}}_{\epsilon}:={{\mathbb P}}\otimes\mu_{\epsilon}$. By ${\mathbb E}_{\epsilon}$ we denote the expectation with respect to ${{\mathbb P}}_{\epsilon}$. From assumptions and we obtain.

\[prop012409\] Under the kinetic scaling we have $$\label{052709-18}
{\cal E}_*(t):=\sup_{{\epsilon}\in(0,1]}\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\|\hat\psi^{({\epsilon})}(t)\|^2_{L^2({{\mathbb T}})}\le
{\cal E}_*+{\gamma}_1 T t,\quad t\ge0.$$

We can introduce the (averaged) Wigner distribution $
W_{\epsilon}(t)\in {\cal A}'$, corresponding to $\psi^{({\epsilon})}(t)$, by the formula $$\label{wigner-bas}
\langle
W_{\epsilon}(t),G\rangle:=\frac{{\epsilon}}{2}\sum_{x,x'\in{{\mathbb Z}}}{\mathbb E}_{\epsilon}\left[\left(\psi^{({\epsilon})}_{x'}(t)\right)^\star
  \psi^{({\epsilon})}_{x}(t)\right]e_{x'-x}(k)G\left({\epsilon}\frac{x+x'}{2},k\right),\quad
G\in {\cal A}.$$ Thanks to it is well defined for any $t\ge0$ and ${\epsilon}\in(0,1]$. Using the Fourier transform in the first variable we can rewrite the Wigner distribution as $$\label{wigner-bas1}
\langle
W_{\epsilon}(t),G\rangle=\int_{{{\mathbb T}}_{\epsilon}\times{{\mathbb T}}}\widehat
W_{\epsilon}^\star(t,\eta,k)\widehat G\left(\eta,k\right)d\eta dk,\quad
G\in {\cal A},$$ where $$\label{W+}
\widehat
W_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\left(\hat\psi^{({\epsilon})}\right)^\star\left(t,k-\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\right].$$ We shall refer to $\widehat
W_{\epsilon}(t)$ as the Fourier-Wigner function corresponding to the given wave function. For the sake of future reference define also $Y_{\epsilon}(t)$, by its Fourier transform $$\label{Y+}
\widehat
Y_{\epsilon}(t,\eta,k):=\frac{{\epsilon}}{2}{\mathbb E}_{\epsilon}\left[\hat\psi^{({\epsilon})}\left(t,-k+\frac{{\epsilon}\eta}{2}\right) \hat\psi^{({\epsilon})}\left(t,k+\frac{{\epsilon}\eta}{2}\right)\right].$$

Kinetic equation
----------------

An important role in our analysis will be played by the function, see Section 2 of [@kors], $$\label{eq:bessel0}
  J(t) = \int_{{{\mathbb T}}}\cos\left(\omega(k) t\right) dk.$$ 