a [*Pinsker type inequality*]{} [@Pi64] for our entropy. Let $\tilde f := h+M_1$. Then $$\label{pinsker}
     \| \tilde f - M_1\|_{L^1(\tilde{\mathbf{T}}\times{\mathbb{R}}^d,{\,\operatorname{d}\!\tilde x}{\,\operatorname{d}\!v})}  \le \sqrt{C_d(L) {\mathcal{E}}^d(\tilde  f)}\ .$$

2.  For any solution $h(t)$ to with ${\mathcal{E}}^d(h^I+M_1)<\infty$, normalized according to , the function $\tilde f(t):=h(t)+M_1$ satisfies $$\label{twoscale1}
    \| \tilde f(t) - M_1\|_{L^1(\tilde{\mathbf{T}}\times{\mathbb{R}}^d,{\,\operatorname{d}\!\tilde x}{\,\operatorname{d}\!v})} \le \sqrt{C_d(L) {\mathcal{E}}^d(\tilde f^I)} e^{-\lambda^d(L)\ t / 2}\,,$$ due to  and  with $\tilde f^I :=h^I +M_1$. However, since $\tilde f(t)$ and $M_1$ are both probability measures, we also have $$\label{twoscale2}
    \| \tilde f(t) - M_1\|_{L^1(\tilde{\mathbf{T}}\times{\mathbb{R}}^d,{\,\operatorname{d}\!\tilde x}{\,\operatorname{d}\!v})}  \le 2$$ for all $t$. Moreover, if most of the mass density is initially located in a small portion of $\tilde{\mathbf{T}}$; e.g., if the gas molecules are initially released from a small container into a vacuum in the rest of $\tilde{\mathbf{T}}$, then $\| \tilde f(t) - M_1\|_{L^1(\tilde{\mathbf{T}}\times{\mathbb{R}}^d,{\,\operatorname{d}\!\tilde x}{\,\operatorname{d}\!v})}$ will be close to $2$ until the streaming has had time to distribute the particles more uniformly over $\tilde {\mathbf{T}}$. Our estimates bound the time that it takes for this to happen.

    Combining with yields $$\label{Hnorm-decay-est}
     \| \tilde f(t) - M_1\|_{L^1(\tilde{\mathbf{T}}\times{\mathbb{R}}^d,{\,\operatorname{d}\!\tilde x}{\,\operatorname{d}\!v})}  \le \min\left\{2,\,\sqrt{C_d(L)\,{\mathcal{E}}^d(\tilde f^I)}\,e^{-\lambda^d(L)\ t/2} \right\}\ ,\qquad t\ge0\ .$$ Our bound  improves the trivial bound  only for $t> t_{{\rm init}}$ where $$t_{{\rm init}} := \frac{ 