mass, zero mean momentum, and unit position-averaged pressure (w.l.o.g. this can be obtained by a Galilean transformation and choice of units). One easily checks that this normalization is conserved under the flow of . Hence the system is expected to have the unique, space-homogeneous steady state $$f^\infty(v) = M_1(v):=(2\pi)^{-\frac{d}2}e^{-\frac{|v|^2}2} \ ,$$ the centered Maxwellian at unit temperature, which clearly has the same normalization as . A standard argument involving the Boltzmann entropy confirms that this is indeed the case, but it gives no information on the rate of convergence to equilibrium, nor does it even prove convergence. We remark that involves two different time scales: the generic transport time is $O(L)$, while the relaxation time is $O(1)$. The goal of this paper is to prove the large time convergence to this $f^\infty$ for solutions of and its linearizations in 1, 2, and 3D with explicitly computable exponential rates.

This extends our previous work [@AAC16], which considered the 1D linear BGK model: $$\label{bgk:linear}
  \partial_t f +v\cdot\nabla_x f = {\mathbf{Q_{lin}}}f := M_{T}(v)\,\int_{\mathbb{R}}f(x,v,t)\,{\,\operatorname{d}\!v} -f(x,v,t)\ ,\quad t\ge0\ ,$$ where $M_{T}$ denotes the normalized Maxwellian at some temperature $T>0$: $$M_T(v) = (2\pi T)^{-1/2}e^{-|v|^2/2T}\ .$$ In [@AAC16] we studied the rate at which normalized solutions of 