absence of the thermostat, it is proven in [@BOS], that as ${\epsilon}\to 0$, the Wigner distribution converges to the solution of the kinetic transport equation $$\label{eq:bos}
  \partial_t W(t,y,k) + \bar\omega'(k) \partial_y W(t,y,k) =
  2\gamma_0 \int_{{{\mathbb T}}} R(k,k') \left(W(t,y,k') - W(t,y,k)\right)  dk, \qquad (t,y,k)\in [0,+\infty)\times{{\mathbb T}}\times \mathbb R,$$ with an explicitly given scattering kernel $ R(k,k') \ge 0$. It is symmetric and the total scattering kernel behaves as $$\label{Rk}
R(k):=\mathlarger{\int}_{{{\mathbb T}}}R(k,k')dk' \sim |k|^2\quad\mbox{ for }|k| \ll 1.$$ Here ${{\mathbb T}}$ is the unit torus, which is the interval $[-1/2,1/2]$, with identified endpoints. Furthermore $\gamma_0>0$ is the scattering rate for the microscopic chain (see below) and $\bar\omega(k) = \omega(k)/2\pi$, where $\omega(k)$ is the dispersion relation of the chain (see definition ).

In the present paper we are interested in the macroscopic effects of a heat bath at temperature $T$, modeled by a Langevin dynamics, applied to one particle, say the one labelled $0$, with a coupling strength $\gamma_1>0$ (see below for a detailed description). Unlike the conservative stochastic dynamics acting on the bulk, the action of the heat bath is not rescaled with ${\epsilon}$, so in the limit as ${\epsilon}\to 0$ it constitutes a singular perturbation on the dynamics. The effect in 