for the microscopic Wigner distributions, see Proposition \[prop012105-19\], that follow from the energy balance equation established in . A similar result is also formulated for the auxiliary dynamics with no stochastic noise in Section \[sec5.1\]. In Section \[sec5.3\] we formulate the result concerning the kinetic limit for this dynamics, see Theorem \[main:thm-un\]. Its proof is quite analogous to the argument of [@kors] and is given in Appendix \[appb\]. Section \[sec5.5\] is essentially devoted to the proof of the main result (Theorem \[main-thm\]) for the case $T=0$ and the proof for $T>0$ is presented in Section \[sec10\]. Some properties of the dynamics corresponding to the macroscopic kinetic limit are proven in Appendix \[appa\]. Section \[appC\] of the appendix is devoted to the proof of some properties of the interface coefficients appearing in the limit.

Acknowledgements {#acknowledgements .unnumbered}
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T.K. acknowledges the support of the National Science Centre: NCN grant DEC-2016/23/B/ST1/00492. S.O. acknowledges the ANR-15-CE40-0020-01 LSD grant of the French National Research Agency.

Preliminaries and statement of the main results {#prelim}
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Basic notation
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We shall use the following notation: let ${{\mathbb R}}_*:={{\mathbb R}}\setminus\{0\}$, ${{\mathbb R}}_+:=(0,+\infty)$, ${{\mathbb R}}_-:=(-\infty,0)$ and likewise ${{\mathbb T}}_*:={{\mathbb T}}\setminus\{0\}$, ${{\mathbb T}}_+:=(0,1/2)$, ${{\mathbb T}}_-:=(-1/2,0)$. Throughout the paper we 