h(x,v,t) = [M_f(x,v,t)  -M_1(v)]  - h(x,v,t)\ , \quad t\geq 0 \ .$$ For $\sigma$, $\mu$, and $\tau$ small we have $$\begin{aligned}
\label{taylor}
 & M_f(x,v) - M_1(v) \nonumber \\
 &= \frac{(1+\sigma)^{1+\frac{d}2}(x)}{\left(2\pi \big\{1+\frac1d\big[\tau(x)-\frac{|\mu|^2(x)}{1+\sigma(x)}\big]\big\}\right)^\frac{d}2} 
 \,\exp\Big\{-\frac{|v(1+\sigma(x))-\mu(x)|^2}{2\big(1+\frac1d\big[\tau(x)-\frac{|\mu|^2(x)}{1+\sigma(x)}\big]\big) (1+\sigma(x))}\Big\}
  - (2\pi)^{-\frac{d}2}e^{-\frac{|v|^2}2}\nonumber \\
  &\approx M_1(v)\left[\big( 1+\frac{d}2 - \frac{|v|^2}{2}\big)\sigma(x)+v\cdot \mu(x)
      + \big( -\frac{1}{2}+ \frac{|v|^2}{2d}\big)\tau(x)\right] \ ,\end{aligned}$$ which yields the linearized BGK model that we shall analyze in dimensions 1, 2, and 3 in this paper: $$\begin{aligned}
 \label{linBGK:torus}
 & \partial_t h (x,v,t) +  v\cdot \nabla_x h(x,v,t) \\
 &\quad = M_1(v)\left[\big( 1+\frac{d}2 - \frac{|v|^2}{2}\big)\sigma(x,t)+v\cdot \mu(x,t)
      + \big( -\frac{1}{2}+ \frac{|v|^2}{2d}\big)\tau(x,t)\right]  - h(x,v,t)\ , \quad t\geq 0 \ .\nonumber\end{aligned}$$ Here and in the sequel we only have $h(x,v,t)\approx f(x,v,t)-M_1(v)$, but for simplicity of notation we shall still denote the perturbation by $h$.

\[linBGK-decay\] For each side length $L>0$ and for dimensions $d=1,2,3$, there exists a (quadratic) entropy functional ${\mathcal{E}}^d(f)$ and a decay rate $\lambda^d(L)>0$ satisfying $$\label{entropy-equiv}
 c_d(L) {\mathcal{E}}^d(f) \leq \| f - M_1\|^2_\mathcal{H} \le C_d(L) {\mathcal{E}}^d(f)\ ,$$ with some positive constants $c_d,\,C_d$ given explicitly in the proofs. Moreover, any solution $h(t)$ to with ${\mathcal{E}}^d(h^I+M_1)<\infty$, normalized according to , then satisfies $$\label{ineq:linBGK-decay}
   {\mathcal{E}}^d(h(t)+M_1) \leq e^{-\lambda^d(L)\ t}\ {\mathcal{E}}^d(h^I+M_1)\ ,\qquad t\ge0\ .
$$

1.  Combining and the bound on the right in , we obtain 