Since $\|f \mathbf{1}_{f \le M}\|_{L_{2\alpha}^1} \le \|\mathcal{M} \mathbf{1}_{f \le M}\|_{L_{2\alpha}^1}$ and, from (92), $\Lambda \le 2$ on $[0, 1]$, we obtain (99) after upper bounding $\|\mathcal{M} \mathbf{1}_{f \le M}\|_{L_{2\alpha}^1}$ by $\|\mathcal{M}\|_{L_{2\alpha}^1}$.

• Proof of (100). We remark that $|f - \mathcal{M}| = f - \mathcal{M} + 2(\mathcal{M} - f)_+$, and

$$ \int_{\mathbb{R}^3} (f - \mathcal{M})(v) (1 + |v|^2) \, dv = 0, $$

so that

$$ \|f - \mathcal{M}\|_{L_2^1}^2 = 4\|(\mathcal{M} - f)_+\|_{L_2^1}^2. $$

We apply (99) with $\alpha = 2$ and obtain (100). □

## APPENDIX A. SIMILAR RESULTS IN THE BOSE-EINSTEIN CASE

An upper-bound inequality similar to (34) can also be obtained in the Bose-Einstein case. This latter case formally corresponds to taking $-\epsilon$ instead of $\epsilon$ in our formulas. First define for any $x \in \mathbb{R}_+$ and $\epsilon > 0$,

$$ \varphi_{\epsilon}^{BE}(x) := \frac{x}{1 + \epsilon x}, \qquad \Phi_{\epsilon}^{BE}(x) := \int_{0}^{x} \log \varphi_{\epsilon}^{BE}(y) \, dy, $$

and the Bose-Einstein entropy of $0 \le f \in L_2^1(\mathbb{R}^3)$:

$$ H_{\epsilon}^{BE}(f) := \int_{\mathbb{R}^3} \Phi_{\epsilon}^{BE}(f) \, dv. $$

Lu proved in [21] that, under the condition

$$ (101) \qquad T \ge \frac{\zeta(\frac{5}{2})}{\zeta(\frac{3}{2})} T_c, \qquad T_c := \frac{1}{2\pi} \left( \frac{\rho \epsilon}{\zeta(\frac{3}{2})} \right)^{2/3}, $$

where $\zeta$ is the Riemann Zêta function and $T_c$ is called the critical temperature, there exists a unique $\epsilon$-Bose-Einstein statistics $\mathcal{M}_{\epsilon}^{BE,f}$ associated to $f$ - that is a distribution such that $\log \varphi_{\epsilon}^{BE}(\mathcal{M}_{\epsilon}^{BE,f})$ is a linear combination of $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$ and sharing the same normalization in $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$ as $f$. We can obtain from Proposition 15 in Appendix B that, denoting $H_{\epsilon}^{BE}[f|\mathcal{M}_{\epsilon}^{BE,f}] := H_{\epsilon}^{BE}(f) - H_{\epsilon}^{BE}(\mathcal{M}_{\epsilon}^{BE,f})$, we have

$$ \begin{align} H_{\epsilon}^{BE}[f|\mathcal{M}_{\epsilon}^{BE,f}] &= \int_{0}^{1} (1-\tau) \left( \int_{\mathbb{R}^3} (f(v) - \mathcal{M}_{\epsilon}^{BE,f}(v))^2 \Phi_{\epsilon}^{BE} \left( (1-\tau)\mathcal{M}_{\epsilon}^{BE,f}(v) + \tau f(v) \right) \, dv \right) \, d\tau \nonumber \\ (102) \qquad \qquad \qquad \qquad &= \int_{\mathbb{R}^3} \int_{\mathcal{M}_{\epsilon}^{BE,f}(v)}^{f(v)} \frac{f(v) - x}{\varphi_{\epsilon}^{BE}(x)} \varphi_{\epsilon}^{BE'}(x) \, dx \, dv, \end{align} $$

where the last equality comes from $\Phi_{\epsilon}^{BE''} = \frac{\varphi_{\epsilon}^{BE'}}{\varphi_{\epsilon}^{BE}}$ and the change of variables $x = \mathcal{M}_{\epsilon}^{BE,f}(v) + \tau(f(v) - \mathcal{M}_{\epsilon}^{BE,f})$.

In the following Proposition, we provide a link between the relative entropies to equilibrium of the Bose-Einstein and the classical cases. Although I believe that both inequalities could be obtained, we only present here the “upper-bound” inequality as its proof is rather short. Further work may allow to obtain the lower-bound inequality, with a constant that probably depends on an $L^\infty$ bound on $f$. This constitutes another reason why we did not investigate this other inequality, as, although $L^\infty$ bounds are natural to use in the Fermi-Dirac context, due to Pauli’s exclusion principle, they are not in the Bose-Einstein one, due to the phenomenon of condensation.

**Proposition 12.** *Upper-bound in the Bose-Einstein case.* For any $\epsilon > 0$ and nonnegative $f \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3) \setminus \{0\}$ which density and temperature satisfy (101), we have

$$ (103) \qquad H_0 \left[ \frac{f}{1 + \epsilon f} \middle| M^{\frac{f}{1 + \epsilon f}} \right] \le H_{\epsilon}^{BE} [f | \mathcal{M}_{\epsilon}^{BE,f}]. $$

*Proof.* The starting point of the proof are Equation (102) and the inequality $|y - z| \ge |\varphi_{\epsilon}^{BE}(y) - \varphi_{\epsilon}^{BE}(z)|$ for all $(y, z) \in \mathbb{R}_+^2$, yielding

$$ H_{\epsilon}^{BE} [f | \mathcal{M}_{\epsilon}^{BE,f}] = \int_{\mathbb{R}^3} \int_{\mathcal{M}_{\epsilon}^{BE,f}}^{f} \frac{f - x}{\varphi_{\epsilon}^{BE}(x)} \varphi_{\epsilon}^{BE'}(x) \, dx \, dv \ge \int_{\mathbb{R}^3} \int_{\mathcal{M}_{\epsilon}^{BE,f}}^{f} \frac{\varphi_{\epsilon}^{BE}(f) - \varphi_{\epsilon}^{BE}(x)}{\varphi_{\epsilon}^{BE}(x)} \varphi_{\epsilon}^{BE'}(x) \, dx \, dv. $$
Since $\|f \mathbf{1}_{f \le M}\|_{L_{2\alpha}^1} \le \|\mathcal{M} \mathbf{1}_{f \le M}\|_{L_{2\alpha}^1}$ and, from (92), $\Lambda \le 2$ on $[0, 1]$, we obtain (99) after upper bounding $\|\mathcal{M} \mathbf{1}_{f \le M}\|_{L_{2\alpha}^1}$ by $\|\mathcal{M}\|_{L_{2\alpha}^1}$.

• Proof of (100). We remark that $|f - \mathcal{M}| = f - \mathcal{M} + 2(\mathcal{M} - f)_+$, and

$$ \int_{\mathbb{R}^3} (f - \mathcal{M})(v) (1 + |v|^2) \, dv = 0, $$

so that

$$ \|f - \mathcal{M}\|_{L_2^1}^2 = 4\|(\mathcal{M} - f)_+\|_{L_2^1}^2. $$

We apply (99) with $\alpha = 2$ and obtain (100). □

## APPENDIX A. SIMILAR RESULTS IN THE BOSE-EINSTEIN CASE

An upper-bound inequality similar to (34) can also be obtained in the Bose-Einstein case. This latter case formally corresponds to taking $-\varepsilon$ instead of $\varepsilon$ in our formulas. First define for any $x \in \mathbb{R}_+$ and $\varepsilon > 0$,

$$ \varphi_{\varepsilon}^{BE}(x) := \frac{x}{1 + \varepsilon x}, \qquad \Phi_{\varepsilon}^{BE}(x) := \int_{0}^{x} \log \varphi_{\varepsilon}^{BE}(y) \, dy, $$

and the Bose-Einstein entropy of $0 \le f \in L_2^1(\mathbb{R}^3)$:

$$ H_{\varepsilon}^{BE}(f) := \int_{\mathbb{R}^3} \Phi_{\varepsilon}^{BE}(f) \, dv. $$

Lu proved in [21] that, under the condition

$$ (101) \qquad T \ge \frac{\zeta(\frac{5}{2})}{\zeta(\frac{3}{2})} T_c, \qquad T_c := \frac{1}{2\pi} \left( \frac{\rho \varepsilon}{\zeta(\frac{3}{2})} \right)^{2/3}, $$

where $\zeta$ is the Riemann Zêta function and $T_c$ is called the critical temperature, there exists a unique $\varepsilon$-Bose-Einstein statistics $\mathcal{M}_{\varepsilon}^{BE,f}$ associated to $f$ - that is a distribution such that $\log \varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})$ is a linear combination of $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$ and sharing the same normalization in $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$ as $f$. We can obtain from Proposition 15 in Appendix B that, denoting $H_{\varepsilon}^{BE}[f|\mathcal{M}_{\varepsilon}^{BE,f}] := H_{\varepsilon}^{BE}(f) - H_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})$, we have

$$ \begin{align} H_{\varepsilon}^{BE}[f|\mathcal{M}_{\varepsilon}^{BE,f}] &= \int_{0}^{1} (1-\tau) \left( \int_{\mathbb{R}^3} (f(v) - \mathcal{M}_{\varepsilon}^{BE,f}(v))^2 \Phi_{\varepsilon}^{BE} \left( (1-\tau)\mathcal{M}_{\varepsilon}^{BE,f}(v) + \tau f(v) \right) \, dv \right) \, d\tau \nonumber \\ (102) \qquad \qquad \qquad \qquad &= \int_{\mathbb{R}^3} \int_{\mathcal{M}_{\varepsilon}^{BE,f}(v)}^{f(v)} \frac{f(v) - x}{\varphi_{\varepsilon}^{BE}(x)} \varphi_{\varepsilon}^{BE'}(x) \, dx \, dv, \end{align} $$

where the last equality comes from $\Phi_{\varepsilon}^{BE''} = \frac{\varphi_{\varepsilon}^{BE'}}{\varphi_{\varepsilon}^{BE}}$ and the change of variables $x = \mathcal{M}_{\varepsilon}^{BE,f}(v) + \tau(f(v) - \mathcal{M}_{\varepsilon}^{BE,f})$.

In the following Proposition, we provide a link between the relative entropies to equilibrium of the Bose-Einstein and the classical cases. Although I believe that both inequalities could be obtained, we only present here the “upper-bound” inequality as its proof is rather short. Further work may allow to obtain the lower-bound inequality, with a constant that probably depends on an $L^\infty$ bound on $f$. This constitutes another reason why we did not investigate this other inequality, as, although $L^\infty$ bounds are natural to use in the Fermi-Dirac context, due to Pauli’s exclusion principle, they are not in the Bose-Einstein one, due to the phenomenon of condensation.

**Proposition 12.** *Upper-bound in the Bose-Einstein case.* For any $\varepsilon > 0$ and nonnegative $f \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3) \setminus \{0\}$ which density and temperature satisfy (101), we have

$$ (103) \qquad H_0 \left[ \frac{f}{1 + \varepsilon f} \middle| M^{\frac{f}{1 + \varepsilon f}} \right] \le H_{\varepsilon}^{BE} [f | \mathcal{M}_{\varepsilon}^{BE,f}]. $$

*Proof.* The starting point of the proof are Equation (102) and the inequality $|y - z| \ge |\varphi_{\varepsilon}^{BE}(y) - \varphi_{\varepsilon}^{BE}(z)|$ for all $(y, z) \in \mathbb{R}_+^2$, yielding

$$ H_{\varepsilon}^{BE} [f | \mathcal{M}_{\varepsilon}^{BE,f}] = \int_{\mathbb{R}^3} \int_{\mathcal{M}_{\varepsilon}^{BE,f}}^{f} \frac{f - x}{\varphi_{\varepsilon}^{BE}(x)} \varphi_{\varepsilon}^{BE'}(x) \, dx \, dv \ge \int_{\mathbb{R}^3} \int_{\mathcal{M}_{\varepsilon}^{BE,f}}^{f} \frac{\varphi_{\varepsilon}^{BE}(f) - \varphi_{\varepsilon}^{BE}(x)}{\varphi_{\varepsilon}^{BE}(x)} \varphi_{\varepsilon}^{BE'}(x) \, dx \, dv. $$
Applying the change of variables $y = \varphi_{\varepsilon}^{BE}(x)$ and using formula (27), we obtain

$$H_{\varepsilon}^{BE}[f|\mathcal{M}_{\varepsilon}^{BE,f}] \geq H_0 \left[ \varphi_{\varepsilon}^{BE}(f) |M^{\varphi_{\varepsilon}^{BE}(f)}| \right] + \int_{\mathbb{R}^3} \int_{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} M^{\varphi_{\varepsilon}^{BE}(f)} \frac{\varphi_{\varepsilon}^{BE}(f) - y}{y} dy \, dv.$$

Remark that, as $\log M^{\varphi_{\varepsilon}^{BE}(f)} - \log \varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})$ is a linear combination of conserved quantities (namely, $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$), we have by definition of $M^{\varphi_{\varepsilon}^{BE}(f)}$ that

$$\int_{\mathbb{R}^3} \varphi_{\varepsilon}^{BE}(f) \log \left( \frac{M^{\varphi_{\varepsilon}^{BE}(f)}}{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} \right) dv = \int_{\mathbb{R}^3} M^{\varphi_{\varepsilon}^{BE}(f)} \log \left( \frac{M^{\varphi_{\varepsilon}^{BE}(f)}}{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} \right) dv,$$

so that

$$\int_{\mathbb{R}^3} \int_{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} M^{\varphi_{\varepsilon}^{BE}(f)} \frac{\varphi_{\varepsilon}^{BE}(f) - y}{y} dy \, dv = \int_{\mathbb{R}^3} \int_{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} M^{\varphi_{\varepsilon}^{BE}(f)} \frac{M^{\varphi_{\varepsilon}^{BE}(f)} - y}{y} dy \, dv \geq 0,$$

ending the proof. □

**Proposition 13. Bose-Einstein CKP inequality.** Let $\varpi : \mathbb{R}^3 \to \mathbb{R}_+$ be measurable, and $r \in [1, 2]$. We recall the definition (88) of the function $\Lambda$,

$$\Lambda(\lambda) := \begin{cases} \frac{2}{(\lambda - 1)^2} & \text{if } \lambda = 1, \\ \frac{\lambda \log \lambda - \lambda + 1}{\lambda \log \lambda - \lambda + 1} & \text{if } \lambda \in \mathbb{R}_+ \setminus \{1\}. \end{cases}$$

Then for any $\varepsilon > 0$ and $0 \le f \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3) \setminus \{0\}$ which density and temperature satisfy (101), assuming that the norms below are finite,

$$(104) \quad \|(f - \mathcal{M}) \varpi\|_{L^r}^2 \le \|\mathcal{M}(1 + \varepsilon\mathcal{M}) \varpi^2\|_{L^{\frac{r}{2-r}}} \Lambda \left( \frac{\|f(1 + \varepsilon f) \varpi^2\|_{L^{\frac{r}{2-r}}}}{\|\mathcal{M}(1 + \varepsilon\mathcal{M}) \varpi^2\|_{L^{\frac{r}{2-r}}}} \right) H_{\varepsilon}^{BE}[f|\mathcal{M}],$$

where we denoted $\mathcal{M} \equiv \mathcal{M}_{\varepsilon}^{BE,f}$ the $\varepsilon$-Bose-Einstein distribution associated to $f$, and $H_{\varepsilon}^{BE}$ is the Bose-Einstein entropy. When $r = 2$, $L^{\frac{r}{2-r}}$ shall be understood as $L^{\infty}$.

*Proof.* The proof is similar to the one of Proposition 10. Let $\varepsilon > 0$.

We apply Corollary 17 with $\Phi(x) \equiv \Phi_{\varepsilon}^{BE}(x) = \int_{0}^{x} \log \frac{y}{1+\varepsilon y} dy$, $J = \mathbb{R}_{+}^{*}$ and

$$\mathcal{F} = \left\{ 0 \le g \in L_2^1(\mathbb{R}^3) \left| \int_{\mathbb{R}^3} g(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho \\ \rho u \\ 3 \rho T + \rho |u|^2 \end{pmatrix} \right. \right\},$$

with $\rho, T$ and $\varepsilon$ such that (101) is satisfied, ensuring the existence of $\mathcal{M}_{\varepsilon}^{BE,f}$. In the rest of this proof, we denote the sake of clarity $\mathcal{M} \equiv \mathcal{M}_{\varepsilon}^{BE,f}$. Then (114) writes, since $H_{\Phi_{\varepsilon}^{BE}} \equiv H_{\varepsilon}^{BE}$ the $\varepsilon$-Bose-Einstein entropy, and $\frac{1}{\Phi_{\varepsilon}^{BE''}}(x) = x(1 + \varepsilon x)$ is convex,

$$\|(f - \mathcal{M}) \varpi\|_{L^r}^2 \le \left( \int_0^1 (1 - \tau) \left\| \left( (1 - \tau)\mathcal{M}(1 + \varepsilon\mathcal{M}) + \tau f(1 + \varepsilon f) \right) \varpi^2 \right\|_{L^{\frac{r}{2-r}}}^{-1} d\tau \right)^{-1} H_{\varepsilon}^{BE}[f|\mathcal{M}].$$

We focus on the term with the integral in $\tau$. From Minkowski's inequality, it is smaller than

$$\left( \int_0^1 \frac{1 - \tau}{(1 - \tau) \|\mathcal{M}(1 + \varepsilon\mathcal{M})\varpi^2\|_{L^{\frac{r}{2-r}}} + \tau \|f(1 + \varepsilon f)\varpi^2\|_{L^{\frac{r}{2-r}}}} d\tau \right)^{-1} \\ = \|\mathcal{M}(1 + \varepsilon\mathcal{M})\varpi^2\|_{L^{\frac{r}{2-r}}} \Lambda \left( \frac{\|f(1 + \varepsilon f)\varpi^2\|_{L^{\frac{r}{2-r}}}}{\|\mathcal{M}(1 + \varepsilon\mathcal{M})\varpi^2\|_{L^{\frac{r}{2-r}}}} \right),$$

where $\Lambda$ is defined in (88) and appears thanks to a Taylor expansion of $\lambda \mapsto \lambda \log \lambda - \lambda$ around 1 like in the proof of Proposition 10, allowing to conclude. □

From the previous proposition, we easily deduce the following standard inequalities.
Applying the change of variables $y = \varphi_{\varepsilon}^{BE}(x)$ and using formula (27), we obtain

$$H_{\varepsilon}^{BE}[f|\mathcal{M}_{\varepsilon}^{BE,f}] \geq H_0 \left[ \varphi_{\varepsilon}^{BE}(f) |M^{\varphi_{\varepsilon}^{BE}(f)}| \right] + \int_{\mathbb{R}^3} \int_{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} M^{\varphi_{\varepsilon}^{BE}(f)} \frac{\varphi_{\varepsilon}^{BE}(f) - y}{y} dy \, dv.$$

Remark that, as $\log M^{\varphi_{\varepsilon}^{BE}(f)} - \log \varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})$ is a linear combination of conserved quantities (namely, $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$), we have by definition of $M^{\varphi_{\varepsilon}^{BE}(f)}$ that

$$\int_{\mathbb{R}^3} \varphi_{\varepsilon}^{BE}(f) \log \left( \frac{M^{\varphi_{\varepsilon}^{BE}(f)}}{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} \right) dv = \int_{\mathbb{R}^3} M^{\varphi_{\varepsilon}^{BE}(f)} \log \left( \frac{M^{\varphi_{\varepsilon}^{BE}(f)}}{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} \right) dv,$$

so that

$$\int_{\mathbb{R}^3} \int_{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} M^{\varphi_{\varepsilon}^{BE}(f)} \frac{\varphi_{\varepsilon}^{BE}(f) - y}{y} dy \, dv = \int_{\mathbb{R}^3} \int_{\varphi_{\varepsilon}^{BE}(\mathcal{M}_{\varepsilon}^{BE,f})} M^{\varphi_{\varepsilon}^{BE}(f)} \frac{M^{\varphi_{\varepsilon}^{BE}(f)} - y}{y} dy \, dv \geq 0,$$

ending the proof.

□

**Proposition 13. Bose-Einstein CKP inequality.** Let $\varpi : \mathbb{R}^3 \to \mathbb{R}_+$ be measurable, and $r \in [1, 2]$. We recall the definition (88) of the function $\Lambda$,

$$\Lambda(\lambda) := \begin{cases} \frac{2}{(\lambda - 1)^2} & \text{if } \lambda = 1, \\ \frac{\lambda \log \lambda - \lambda + 1}{\lambda \log \lambda - \lambda + 1} & \text{if } \lambda \in \mathbb{R}_+ \setminus \{1\}. \end{cases}$$

Then for any $\varepsilon > 0$ and $0 \le f \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3) \setminus \{0\}$ which density and temperature satisfy (101), assuming that the norms below are finite,

$$(104) \quad \|(f - \mathcal{M}) \varpi\|_{L^r}^2 \le \|\mathcal{M}(1 + \varepsilon\mathcal{M}) \varpi^2\|_{L^{\frac{r}{2-r}}} \Lambda \left( \frac{\|f(1 + \varepsilon f) \varpi^2\|_{L^{\frac{r}{2-r}}}}{\|\mathcal{M}(1 + \varepsilon\mathcal{M}) \varpi^2\|_{L^{\frac{r}{2-r}}}} \right) H_{\varepsilon}^{BE}[f|\mathcal{M}],$$

where we denoted $\mathcal{M} \equiv \mathcal{M}_{\varepsilon}^{BE,f}$ the $\varepsilon$-Bose-Einstein distribution associated to $f$, and $H_{\varepsilon}^{BE}$ is the Bose-Einstein entropy. When $r = 2$, $L^{\frac{r}{2-r}}$ shall be understood as $L^{\infty}$.

*Proof.* The proof is similar to the one of Proposition 10. Let $\varepsilon > 0$.

We apply Corollary 17 with $\Phi(x) \equiv \Phi_{\varepsilon}^{BE}(x) = \int_{0}^{x} \log \frac{y}{1+\varepsilon y} dy$, $J = \mathbb{R}_{+}^{*}$ and

$$\mathcal{F} = \left\{ 0 \le g \in L_2^1(\mathbb{R}^3) \left| \int_{\mathbb{R}^3} g(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho \\ \rho u \\ 3 \rho T + \rho |u|^2 \end{pmatrix} \right. \right\},$$

with $\rho, T$ and $\varepsilon$ such that (101) is satisfied, ensuring the existence of $\mathcal{M}_{\varepsilon}^{BE,f}$. In the rest of this proof, we denote the sake of clarity $\mathcal{M} \equiv \mathcal{M}_{\varepsilon}^{BE,f}$. Then (114) writes, since $H_{\Phi_{\varepsilon}^{BE}} \equiv H_{\varepsilon}^{BE}$ the $\varepsilon$-Bose-Einstein entropy, and $\frac{1}{\Phi_{\varepsilon}^{BE''}}(x) = x(1 + \varepsilon x)$ is convex,

$$\|(f - \mathcal{M}) \varpi\|_{L^r}^2 \le \left( \int_0^1 (1 - \tau) \left\| \left( (1 - \tau)\mathcal{M}(1 + \varepsilon\mathcal{M}) + \tau f(1 + \varepsilon f) \right) \varpi^2 \right\|_{L^{\frac{r}{2-r}}}^{-1} d\tau \right)^{-1} H_{\varepsilon}^{BE}[f|\mathcal{M}].$$

We focus on the term with the integral in $\tau$. From Minkowski's inequality, it is smaller than

$$\left( \int_0^1 \frac{1 - \tau}{(1 - \tau) \|\mathcal{M}(1 + \varepsilon\mathcal{M})\varpi^2\|_{L^{\frac{r}{2-r}}} + \tau \|f(1 + \varepsilon f)\varpi^2\|_{L^{\frac{r}{2-r}}}} d\tau \right)^{-1} \\ = \|\mathcal{M}(1 + \varepsilon\mathcal{M})\varpi^2\|_{L^{\frac{r}{2-r}}} \Lambda \left( \frac{\|f(1 + \varepsilon f)\varpi^2\|_{L^{\frac{r}{2-r}}}}{\|\mathcal{M}(1 + \varepsilon\mathcal{M})\varpi^2\|_{L^{\frac{r}{2-r}}}} \right),$$

where $\Lambda$ is defined in (88) and appears thanks to a Taylor expansion of $\lambda \mapsto \lambda \log \lambda - \lambda$ around 1 like in the proof of Proposition 10, allowing to conclude. □

From the previous proposition, we easily deduce the following standard inequalities.
**Corollary 14.** *Standard Bose-Einstein CKP inequalities.* For any $\varepsilon > 0$, $\alpha \ge 0$ and $0 \le f \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3) \setminus \{0\}$ satisfying (101),

$$ (105) \qquad \|(M - f)_+\|_{L_\alpha^1}^2 \le 2 (\|M\|_{L_{2\alpha}^1} + \varepsilon \|M\|_{L_\alpha^2}^2) H_\varepsilon^{BE}[f|M], $$

$$ (106) \qquad \|f - M\|_{L_1^1}^2 \le 8 (\|M\|_{L_1^1} + \varepsilon \|M\|_{L_2^2}^2) H_\varepsilon^{BE}[f|M], $$

$$ (107) \qquad \|f - M\|_{L_2^1}^2 \le 8 (\|M\|_{L_4^1} + \varepsilon \|M\|_{L_2^2}^2) H_\varepsilon^{BE}[f|M], $$

where we denoted for clarity $M \equiv M_\varepsilon^{BE,f}$ the $\varepsilon$-Bose-Einstein distribution associated to $f$.

We prove the above inequalities similarly as we did for Corollary 11. For (105) we apply Proposition 13 with $r = 1$ and $\varpi(v) = (1 + |v|^2)^{\frac{\alpha}{2}} \mathbf{1}_{f \le M}$ and notice that $\Lambda \le 2$ on $[0, 1]$. We then obtain (106)-(107) by decomposing $|f - M| = f - M + 2(M - f)_+$, using the fact that $f$ and $M$ share the same normalization in $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$, and (105) with respectively $\alpha = 0$ and $\alpha = 2$.

## APPENDIX B. A GENERAL DISCUSSION ABOUT ENTROPIES AND EQUILIBRIA

In this section we intend to provide general considerations on the entropy, which are much more general than the scope of this paper, but give a good understanding of the notions we used, and could also be helpful in the study of weak turbulence, where various kinds of unusual entropies can emerge (see [7]). Our setting is laid down quite generally. Consider a measured space $(\mathcal{E}, \mathcal{A}, \mu)$, an open interval $J \subset \mathbb{R}$ which closure we denote by $\bar{J}$, and a function $\Phi \in C^2(J) \cap C^0(\bar{J})$ such that $\Phi'' > 0$ on $J$. Remark that $\Phi'$ is then a $C^1$-diffeomorphism from $J$ onto $\Phi'(J)$.

**Entropy.** We define the $\Phi$-entropy, for any $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $f : \mathcal{E} \to \bar{J}$ such that the following integral makes sense and is finite, by

$$ (108) \qquad H_{\Phi}(f) := \int_{\mathcal{E}} \Phi(f(\zeta)) \, d\mu(\zeta), $$

and we denote by $E_\Phi$ the set of such $f$. We let the relative $\Phi$-entropy of $f$ and $g$ to be

$$ (109) \qquad H_{\Phi}[f|g] = H_{\Phi}(f) - H_{\Phi}(g). $$

We also define, for $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $f, g : \mathcal{E} \to \bar{J}$, the $\Phi$-relative-entropy (which in general differs from the relative $\Phi$-entropy) of $f$ and $g$ by

$$ (110) \qquad \mathcal{H}_{\Phi}[f|g] := \int_{0}^{1} (1 - \tau) \left( \int_{g \in J} (f - g)^2 \Phi''((1 - \tau)g + \tau f) \, d\mu(\zeta) \right) \, d\tau. $$

Note that $\mathcal{H}_{\Phi}[f|g]$ is possibly infinite, but always well-defined, as

$$ g \in J \implies \forall \tau \in (0, 1), \quad (1 - \tau)g + \tau f \in J, $$

and that $\mathcal{H}_{\Phi}[f|g]$ is always nonnegative. The following Proposition 15 gives a quite simple but general result linking entropies, conserved quantities and equilibrium distributions, under a sole existence assumption.

**Proposition 15.** Let $I$ be a countable set, $(\phi_i)_{i \in I}$ a family of measurable real functions, $(\omega_i)_{i \in I}$ a family of real numbers, and

$$ \mathcal{F} = \left\{ f \in E_{\Phi} \text{ s.t. } \forall i \in I, \int_{\mathcal{E}} |f(\zeta)| |\phi_i(\zeta)| \, d\mu(\zeta) < \infty \text{ and } \int_{\mathcal{E}} f(\zeta) \phi_i(\zeta) \, d\mu(\zeta) = \omega_i \right\}. $$

Assume that $\Phi'(J) = \mathbb{R}$, and that there exists $(\alpha_i(\omega)) \in \mathbb{R}^I$ such that $M_{\Phi}^{\mathcal{F}} \in \mathcal{F}$, where

$$ (111) \qquad M_{\Phi}^{\mathcal{F}} := (\Phi')^{-1} \left( \sum_{i \in I} \alpha_i(\omega) \phi_i \right). $$
**Corollary 14.** *Standard Bose-Einstein CKP inequalities.* For any $\varepsilon > 0$, $\alpha \ge 0$ and $0 \le f \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3) \setminus \{0\}$ satisfying (101),

$$ (105) \qquad \|(M - f)_+\|_{L_\alpha^1}^2 \le 2 \left( \|M\|_{L_{2\alpha}^1} + \varepsilon \|M\|_{L_\alpha^2}^2 \right) H_\varepsilon^{BE}[f|M], $$

$$ (106) \qquad \|f - M\|_{L_1^1}^2 \le 8 \left( \|M\|_{L_1^1} + \varepsilon \|M\|_{L_2^2}^2 \right) H_\varepsilon^{BE}[f|M], $$

$$ (107) \qquad \|f - M\|_{L_2^1}^2 \le 8 \left( \|M\|_{L_4^1} + \varepsilon \|M\|_{L_2^2}^2 \right) H_\varepsilon^{BE}[f|M], $$

where we denoted for clarity $M \equiv M_\varepsilon^{BE,f}$ the $\varepsilon$-Bose-Einstein distribution associated to $f$.

We prove the above inequalities similarly as we did for Corollary 11. For (105) we apply Proposition 13 with $r = 1$ and $\varpi(v) = (1 + |v|^2)^{\frac{\alpha}{2}} \mathbf{1}_{f \le M}$ and notice that $\Lambda \le 2$ on $[0, 1]$. We then obtain (106)-(107) by decomposing $|f - M| = f - M + 2(M - f)_+$, using the fact that $f$ and $M$ share the same normalization in $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$, and (105) with respectively $\alpha = 0$ and $\alpha = 2$.

## APPENDIX B. A GENERAL DISCUSSION ABOUT ENTROPIES AND EQUILIBRIA

In this section we intend to provide general considerations on the entropy, which are much more general than the scope of this paper, but give a good understanding of the notions we used, and could also be helpful in the study of weak turbulence, where various kinds of unusual entropies can emerge (see [7]). Our setting is laid down quite generally. Consider a measured space $(\mathcal{E}, \mathcal{A}, \mu)$, an open interval $J \subset \mathbb{R}$ which closure we denote by $\bar{J}$, and a function $\Phi \in C^2(J) \cap C^0(\bar{J})$ such that $\Phi'' > 0$ on $J$. Remark that $\Phi'$ is then a $C^1$-diffeomorphism from $J$ onto $\Phi'(J)$.

**Entropy.** We define the $\Phi$-entropy, for any $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $f : \mathcal{E} \to \bar{J}$ such that the following integral makes sense and is finite, by

$$ (108) \qquad H_{\Phi}(f) := \int_{\mathcal{E}} \Phi(f(\zeta)) \, d\mu(\zeta), $$

and we denote by $E_\Phi$ the set of such $f$. We let the relative $\Phi$-entropy of $f$ and $g$ to be

$$ (109) \qquad H_{\Phi}[f|g] = H_{\Phi}(f) - H_{\Phi}(g). $$

We also define, for $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $f, g : \mathcal{E} \to \bar{J}$, the $\Phi$-relative-entropy (which in general differs from the relative $\Phi$-entropy) of $f$ and $g$ by

$$ (110) \qquad \mathcal{H}_{\Phi}[f|g] := \int_{0}^{1} (1 - \tau) \left( \int_{g \in J} (f - g)^2 \Phi''((1 - \tau)g + \tau f) \, d\mu(\zeta) \right) \, d\tau. $$

Note that $\mathcal{H}_{\Phi}[f|g]$ is possibly infinite, but always well-defined, as

$$ g \in J \implies \forall \tau \in (0, 1), \quad (1 - \tau)g + \tau f \in J, $$

and that $\mathcal{H}_{\Phi}[f|g]$ is always nonnegative. The following Proposition 15 gives a quite simple but general result linking entropies, conserved quantities and equilibrium distributions, under a sole existence assumption.

**Proposition 15.** Let $I$ be a countable set, $(\phi_i)_{i \in I}$ a family of measurable real functions, $(\omega_i)_{i \in I}$ a family of real numbers, and

$$ \mathcal{F} = \left\{ f \in E_{\Phi} \text{ s.t. } \forall i \in I, \int_{\mathcal{E}} |f(\zeta)| |\phi_i(\zeta)| \, d\mu(\zeta) < \infty \text{ and } \int_{\mathcal{E}} f(\zeta) \phi_i(\zeta) \, d\mu(\zeta) = \omega_i \right\}. $$

Assume that $\Phi'(J) = \mathbb{R}$, and that there exists $(\alpha_i(\omega)) \in \mathbb{R}^I$ such that $M_{\Phi}^{\mathcal{F}} \in \mathcal{F}$, where

$$ (111) \qquad M_{\Phi}^{\mathcal{F}} := (\Phi')^{-1} \left( \sum_{i \in I} \alpha_i(\omega) \phi_i \right). $$
Then the following four propositions are equivalent. Let $g \in \mathcal{F}$.

$$ (i) \ g \in J \ \mu\text{-a.e. and} \ \forall f \in \mathcal{F}, \ H_{\Phi}[f|g] = \mathcal{H}_{\Phi}[f|g], $$

$$ (ii) \ g \in J \ \mu\text{-a.e. and} \ \forall f \in \mathcal{F}, \ \mathcal{H}_{\Phi}[f|g] < \infty \ \text{and} \ \int_{\mathcal{E}} (f-g) \Phi'(g) \, d\mu(\zeta) = 0, $$

$$ (iii) \ H_{\Phi}(g) = \min_{h \in \mathcal{F}} H_{\Phi}(h), $$

$$ (iv) \ g = M_{\Phi}^{\mathcal{F}} \ \mu\text{-a.e.} $$

In particular, $M_{\Phi}^{\mathcal{F}}$ is the unique minimizer of $H_{\Phi}$ under the constraints of the set $\mathcal{F}$.

The above proposition actually proves the following (under the assumptions $\Phi'(J) = \mathbb{R}$ and of existence of $M_{\Phi}^{\mathcal{F}}$). An admissible distribution $g$ is an equilibrium relative to the conserved quantities $\phi_i$, in the sense of the minimization of the $\Phi$-entropy [(iii)], if and only if $\Phi'(g)$ is a linear combination of the functions $\phi_i$ [(iv)], if and only if the relative $\Phi$-entropy between any admissible distribution $f$ and $g$ is given by (110) [(i)], if and only if the quantity $\Phi'(g)$ is conserved amongst all admissible distributions [(ii)] - indeed, (ii) is equivalent, assuming the following integrals make sense, to $\forall f_1, f_2 \in \mathcal{F}, \int_{\mathcal{E}} f_1 \Phi'(g) d\mu(\zeta) = \int_{\mathcal{E}} f_2 \Phi'(g) d\mu(\zeta)$.

The reader may notice that the Classical case corresponds to the choice $\Phi(x) = \Phi_0(x) = x \log x - x$, for which $(\Phi'_0)^{-1} = \exp$, hence $M_{\Phi}^{\mathcal{F}}$ is in this case a Maxwellian, since the conserved quantities (corresponding to the functions $\phi_i$ in the proposition) are $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$.

Moreover, the Fermi-Dirac case corresponds to the choice $\Phi(x) = \Phi_{\varepsilon}(x) = \int_{0}^{x} \log \frac{y}{1-\varepsilon y} dy$, for which $(\Phi'_{\varepsilon})^{-1}(x) = \frac{e^x}{1+\varepsilon e^x}$, hence $M_{\Phi}^{\mathcal{F}}$ is in this case a Fermi-Dirac distribution, since again, the conserved quantities (corresponding to the functions $\phi_i$ in the proposition) are $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$. Proposition 1 then comes as a corollary to Proposition 15 with $\mathcal{E} = \mathbb{R}^3$ endowed with the Lebesgue measure, $J = (0, \varepsilon^{-1})$, $\Phi(x) = \Phi_{\varepsilon}(x) = \int_{0}^{x} \log \varphi_{\varepsilon}(y) dy$, $M_{\Phi}^{\mathcal{F}} = \mathcal{M}_{\varepsilon}^f$ and

$$ \mathcal{F} = \left\{ 0 \le f \in L_2^1(\mathbb{R}^3) \mid 1 - \varepsilon f \ge 0, \quad \int_{\mathbb{R}^3} f(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho \\ \rho u \\ 3 \rho T + \rho |u|^2 \end{pmatrix} \right\}, $$

with $\rho, T$ and $\varepsilon$ such that $\gamma > \frac{2}{5}$ (ensuring the existence of $\mathcal{M}_{\varepsilon}^f$, as proven in [22]).

**Remark 6.** The Bose-Einstein case is also recovered with $\Phi(x) = \Phi_{\varepsilon}^{BE}(x) = \int_{0}^{x} \log \frac{y}{1+\varepsilon y} dy$, for which $(\Phi'_{-\varepsilon})^{-1}(x) = \frac{e^x}{1-\varepsilon e^x}$, hence $M_{\Phi}^{\mathcal{F}}$ is in this case a Bose-Einstein distribution (when it exists).

*Proof.* We start by proving (i) $\iff$ (ii). Let $f, g \in \mathcal{F}$ such that $g \in J$ $\mu$-almost everywhere and $\mathcal{H}_{\Phi}[f|g] < \infty$. Using a Taylor expansion followed by Fubini's Theorem, we get

$$ \begin{aligned} H_{\Phi}[f|g] &= \int_{\mathcal{E}} (\Phi(f) - \Phi(g)) d\mu(\zeta) = \int_{g \in J} (\Phi(f) - \Phi(g)) d\mu(\zeta) \\ &= \int_{g \in J} \left( (f-g)\Phi'(g) + (f-g)^2 \int_{0}^{1} (1-\tau)\Phi''((1-\tau)g + \tau f) d\tau \right) d\mu(\zeta) \\ &= \int_{g \in J} (f-g)\Phi'(g) d\mu(\zeta) + \mathcal{H}_{\Phi}[f|g], \end{aligned} $$

thus proving the announced equivalence. Remark that in the last equality we used the fact that $\mathcal{H}_{\Phi}[f|g] < \infty$ to ensure that $\int_{g \in J} (f-g)\Phi'(g) d\mu(\zeta)$ is well-defined. We now show (iv) $\implies$ (ii). Since $\mathrm{Im}(\Phi'^{-1}) = J$, we
Then the following four propositions are equivalent. Let $g \in \mathcal{F}$.

(i) $g \in J$ $\mu$-a.e. and $\forall f \in \mathcal{F}$, $H_{\Phi}[f|g] = \mathcal{H}_{\Phi}[f|g]$,

(ii) $g \in J$ $\mu$-a.e. and $\forall f \in \mathcal{F}$, $\mathcal{H}_{\Phi}[f|g] < \infty$ and $\int_{\mathcal{E}} (f-g) \Phi'(g) d\mu(\zeta) = 0$,

$$
(iii) H_{\Phi}(g) = \min_{h \in \mathcal{F}} H_{\Phi}(h),
$$

$$
(iv) g = M_{\Phi}^{\mathcal{F}} \quad \mu\text{-a.e.}
$$

In particular, $M_{\Phi}^{\mathcal{F}}$ is the unique minimizer of $H_{\Phi}$ under the constraints of the set $\mathcal{F}$.

The above proposition actually proves the following (under the assumptions $\Phi'(J) = \mathbb{R}$ and of existence of $M_{\Phi}^{\mathcal{F}}$). An admissible distribution $g$ is an equilibrium relative to the conserved quantities $\phi_i$, in the sense of the minimization of the $\Phi$-entropy [(iii)], if and only if $\Phi'(g)$ is a linear combination of the functions $\phi_i$ [(iv)], if and only if the relative $\Phi$-entropy between any admissible distribution $f$ and $g$ is given by (110) [(i)], if and only if the quantity $\Phi'(g)$ is conserved amongst all admissible distributions [(ii)] - indeed, (ii) is equivalent, assuming the following integrals make sense, to $\forall f_1, f_2 \in \mathcal{F}, \int_{\mathcal{E}} f_1 \Phi'(g) d\mu(\zeta) = \int_{\mathcal{E}} f_2 \Phi'(g) d\mu(\zeta)$.

The reader may notice that the Classical case corresponds to the choice $\Phi(x) \equiv \Phi_0(x) = x \log x - x$, for which $(\Phi'_0)^{-1} = \exp$, hence $M_{\Phi}^{\mathcal{F}}$ is in this case a Maxwellian, since the conserved quantities (corresponding to the functions $\phi_i$ in the proposition) are $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$.

Moreover, the Fermi-Dirac case corresponds to the choice $\Phi(x) \equiv \Phi_{\varepsilon}(x) \equiv \int_{0}^{x} \log \frac{y}{1-\varepsilon y} dy$, for which $(\Phi'_{\varepsilon})^{-1}(x) = \frac{e^x}{1+\varepsilon e^x}$, hence $M_{\Phi}^{\mathcal{F}}$ is in this case a Fermi-Dirac distribution, since again, the conserved quantities (corresponding to the functions $\phi_i$ in the proposition) are $v \mapsto 1$, $v \mapsto v$ and $v \mapsto |v|^2$. Proposition 1 then comes as a corollary to Proposition 15 with $\mathcal{E} = \mathbb{R}^3$ endowed with the Lebesgue measure, $J = (0, \varepsilon^{-1})$, $\Phi(x) \equiv \Phi_{\varepsilon}(x) \equiv \int_{0}^{x} \log \varphi_{\varepsilon}(y) dy$, $M_{\Phi}^{\mathcal{F}} \equiv \mathcal{M}_{\varepsilon}^{f}$ and

$$
\mathcal{F} = \left\{ 0 \le f \in L_2^1(\mathbb{R}^3) \left| \ 1 - \varepsilon f \ge 0, \quad \int_{\mathbb{R}^3} f(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho \\ \rho u \\ 3 \rho T + \rho |u|^2 \end{pmatrix} \right\},
$$

with $\rho, T$ and $\varepsilon$ such that $\gamma > \frac{2}{5}$ (ensuring the existence of $M_{\varepsilon}^f$, as proven in [22]).

**Remark 6.** The Bose-Einstein case is also recovered with $\Phi(x) \equiv \Phi_{\varepsilon}^{BE}(x) \equiv \int_{0}^{x} \log \frac{y}{1+\varepsilon y} dy$, for which $(\Phi'_{-\varepsilon})^{-1}(x) = \frac{e^x}{1-\varepsilon e^x}$, hence $M_{\Phi}^{\mathcal{F}}$ is in this case a Bose-Einstein distribution (when it exists).

*Proof.* We start by proving (i) ⇐⇒ (ii). Let $f, g \in \mathcal{F}$ such that $g \in J$ $\mu$-almost everywhere and $\mathcal{H}_{\Phi}[f|g] < \infty$. Using a Taylor expansion followed by Fubini's Theorem, we get

$$
\begin{align*}
H_{\Phi}[f|g] &= \int_{\mathcal{E}} (\Phi(f) - \Phi(g)) \, d\mu(\zeta) = \int_{g \in J} (\Phi(f) - \Phi(g)) \, d\mu(\zeta) \\
&= \int_{g \in J} \left( (f-g)\Phi'(g) + (f-g)^2 \int_{0}^{1} (1-\tau)\Phi''((1-\tau)g + \tau f) \, d\tau \right) \, d\mu(\zeta) \\
&= \int_{g \in J} (f-g)\Phi'(g) \, d\mu(\zeta) + \mathcal{H}_{\Phi}[f|g],
\end{align*}
$$

thus proving the announced equivalence. Remark that in the last equality we used the fact that $\mathcal{H}_{\Phi}[f|g] < \infty$
to ensure that $\int_{g \in J} (f-g)\Phi'(g) d\mu(\zeta)$ is well-defined. We now show (iv) ⇒ (ii). Since $\mathrm{Im}(\Phi'^{-1}) = J$, we
do have $M_{\Phi}^{\mathcal{F}} \in J$ $\mu$-almost everywhere, and

$$
\begin{align*}
\int_{M_{\Phi}^{\mathcal{F}} \in J} (f - M_{\Phi}^{\mathcal{F}}) \Phi'(M_{\Phi}^{\mathcal{F}}) \, d\mu(\zeta) &= \int_{\mathcal{E}} (f - M_{\Phi}^{\mathcal{F}}) \sum_{i} \alpha_i(\omega_i) \phi_i \, d\mu(\zeta) \\
&= \sum_{i} \alpha_i(\omega_i) \left( \int_{\mathcal{E}} f \phi_i \, d\mu(\zeta) - \int_{\mathcal{E}} M_{\Phi}^{\mathcal{F}} \phi_i \, d\mu(\zeta) \right) = 0,
\end{align*}
$$

where the last equality comes from the fact that both $f$ and $M_{\Phi}^{\mathcal{F}}$ belong to $\mathcal{F}$. We now focus on (ii) $\implies$ (iv).
Assume the existence of $g \in \mathcal{F}$ such that $g \in J$ $\mu$-almost everywhere and

$$
\forall f \in \mathcal{F}, \quad \int_{g \in J} (f - g) \Phi'(g) \, d\mu(\zeta) = 0.
$$

Since we also have $M_{\Phi}^{\mathcal{F}} \in \mathcal{F}$, then $M_{\Phi}^{\mathcal{F}} \in J$ $\mu$-almost everywhere and we just proved that

$$
\int_{M_{\Phi}^{\mathcal{F}} \in J} (g - M_{\Phi}^{\mathcal{F}}) \Phi'(M_{\Phi}^{\mathcal{F}}) \, d\mu(\zeta) = 0,
$$

allowing to deduce that

$$
\int_{\mathcal{E}} (g - M_{\Phi}^{\mathcal{F}})(\Phi'(g) - \Phi'(M_{\Phi}^{\mathcal{F}})) \, d\mu(\zeta) = 0.
$$

Since $\Phi'$ is increasing, this implies that $g = M_{\Phi}^{\mathcal{F}}$ $\mu$-almost everywhere. We now focus on (iv) $\implies$ (iii).
Since we already proved (iv) $\implies$ (ii) $\implies$ (i), we have for any $f \in \mathcal{F}$,

$$
H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}] = H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}] \geq 0,
$$

thus

$$
H_{\Phi}(f) \geq H_{\Phi}(M_{\Phi}^{\mathcal{F}}).
$$

Finally, we prove (iii) $\implies$ (iv). Assume $H_{\Phi}(g) = \min_{h \in \mathcal{F}} H_{\Phi}(h)$. We just proved that $H_{\Phi}(M_{\Phi}^{\mathcal{F}}) = \min_{h \in \mathcal{F}} H_{\Phi}(h)$,
hence $H_{\Phi}(g) = H_{\Phi}(M_{\Phi}^{\mathcal{F}})$ and $H_{\Phi}[g|M_{\Phi}^{\mathcal{F}}] = 0$. Since we also proved (iv) $\implies$ (i), we know that

$$
\mathcal{H}_{\Phi}[g|M_{\Phi}^{\mathcal{F}}] = H_{\Phi}[g|M_{\Phi}^{\mathcal{F}}],
$$

hence $\mathcal{H}_{\Phi}[g|M_{\Phi}^{\mathcal{F}}] = 0$, that is

$$
\int_0^1 (1 - \tau) \left( \int_{M_{\Phi}^{\mathcal{F}} \in J} (g - M_{\Phi}^{\mathcal{F}})^2 \Phi''((1 - \tau)M + \tau g) \, d\mu(\zeta) \right) d\tau = 0.
$$

This implies, since $M_{\Phi}^{\mathcal{F}} \in J$ and $(1-\tau)M_{\Phi}^{\mathcal{F}} + \tau g \in J$ $\mu$-almost everywhere for any $\tau \in (0,1)$, and $\Phi'' > 0$ on $J$, that $g = M_{\Phi}^{\mathcal{F}}$ $\mu$-almost everywhere. $\square$

**General Csiszár-Kullback-Pinsker inequality.** The famous Csiszár-Kullback-Pinsker inequality, linking the squared $L^1$ distance of two probabilities with their relative Classical entropy can in fact be generalized to the whole family of $\Phi$-entropies, and for weighted $L^r$, $1 \le r \le 2$ distances, by the following Proposition 16, and more specifically, in the context of convergence towards equilibrium, by Corollary 17. Again, such inequalities may be useful in the study of weak turbulence [7].

**Proposition 16.** Let some $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $\varpi : \mathcal{E} \to \bar{J}$ and $r \in [1, 2]$. Then for any $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $f, g : \mathcal{E} \to \bar{J}$ such that all terms below are finite, we have

$$
(112) \quad \|(f-g) \varpi\|_{L^r(g \in J)}^2 \le \left( \int_0^1 (1-\tau) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right)^{-1} \mathcal{H}_{\Phi}[f|g],
$$

where $\mathcal{H}_{\Phi}$ is the $\Phi$-relative-entropy defined in (110).

*Proof.* Let us recall the definition of $\mathcal{H}_{\Phi}[f|g]$, that is

$$
\mathcal{H}_{\Phi}[f|g] = \int_{0}^{1} (1-\tau) \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g + \tau f) \, d\mu(\zeta) \right) d\tau.
$$
do have $M_{\Phi}^{\mathcal{F}} \in J$ $\mu$-almost everywhere, and

$$
\begin{align*}
\int_{M_{\Phi}^{\mathcal{F}} \in J} (f - M_{\Phi}^{\mathcal{F}}) \Phi'(M_{\Phi}^{\mathcal{F}}) \, d\mu(\zeta) &= \int_{\mathcal{E}} (f - M_{\Phi}^{\mathcal{F}}) \sum_{i} \alpha_i(\omega_i) \phi_i \, d\mu(\zeta) \\
&= \sum_{i} \alpha_i(\omega_i) \left( \int_{\mathcal{E}} f \phi_i \, d\mu(\zeta) - \int_{\mathcal{E}} M_{\Phi}^{\mathcal{F}} \phi_i \, d\mu(\zeta) \right) = 0,
\end{align*}
$$

where the last equality comes from the fact that both $f$ and $M_{\Phi}^{\mathcal{F}}$ belong to $\mathcal{F}$. We now focus on (ii) $\implies$ (iv).
Assume the existence of $g \in \mathcal{F}$ such that $g \in J$ $\mu$-almost everywhere and

$$
\forall f \in \mathcal{F}, \quad \int_{g \in J} (f - g) \Phi'(g) \, d\mu(\zeta) = 0.
$$

Since we also have $M_{\Phi}^{\mathcal{F}} \in \mathcal{F}$, then $M_{\Phi}^{\mathcal{F}} \in J$ $\mu$-almost everywhere and we just proved that

$$
\int_{M_{\Phi}^{\mathcal{F}} \in J} (g - M_{\Phi}^{\mathcal{F}}) \Phi'(M_{\Phi}^{\mathcal{F}}) \, d\mu(\zeta) = 0,
$$

allowing to deduce that

$$
\int_{\mathcal{E}} (g - M_{\Phi}^{\mathcal{F}})(\Phi'(g) - \Phi'(M_{\Phi}^{\mathcal{F}})) \, d\mu(\zeta) = 0.
$$

Since $\Phi'$ is increasing, this implies that $g = M_{\Phi}^{\mathcal{F}}$ $\mu$-almost everywhere. We now focus on (iv) $\implies$ (iii).
Since we already proved (iv) $\implies$ (ii) $\implies$ (i), we have for any $f \in \mathcal{F}$,

$$
H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}] = H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}] \geq 0,
$$

thus

$$
H_{\Phi}(f) \geq H_{\Phi}(M_{\Phi}^{\mathcal{F}}).
$$

Finally, we prove (iii) $\implies$ (iv). Assume $H_{\Phi}(g) = \min_{h \in \mathcal{F}} H_{\Phi}(h)$. We just proved that $H_{\Phi}(M_{\Phi}^{\mathcal{F}}) = \min_{h \in \mathcal{F}} H_{\Phi}(h)$,
hence $H_{\Phi}(g) = H_{\Phi}(M_{\Phi}^{\mathcal{F}})$ and $H_{\Phi}[g|M_{\Phi}^{\mathcal{F}}] = 0$. Since we also proved (iv) $\implies$ (i), we know that

$$
\mathcal{H}_{\Phi}[g|M_{\Phi}^{\mathcal{F}}] = H_{\Phi}[g|M_{\Phi}^{\mathcal{F}}],
$$

hence $\mathcal{H}_{\Phi}[g|M_{\Phi}^{\mathcal{F}}] = 0$, that is

$$
\int_0^1 (1-\tau) \left( \int_{M_{\Phi}^{\mathcal{F}} \in J} (g - M_{\Phi}^{\mathcal{F}})^2 \Phi''((1-\tau)M + \tau g) \, d\mu(\zeta) \right)^{-1} d\tau = 0.
$$

This implies, since $M_{\Phi}^{\mathcal{F}} \in J$ and $(1-\tau)M_{\Phi}^{\mathcal{F}} + \tau g \in J$ $\mu$-almost everywhere for any $\tau \in (0,1)$, and $\Phi'' > 0$ on $J$, that $g = M_{\Phi}^{\mathcal{F}}$ $\mu$-almost everywhere. $\square$

**General Csiszár-Kullback-Pinsker inequality.** The famous Csiszár-Kullback-Pinsker inequality, linking the squared $L^1$ distance of two probabilities with their relative Classical entropy can in fact be generalized to the whole family of $\Phi$-entropies, and for weighted $L^r$, $1 \le r \le 2$ distances, by the following Proposition 16, and more specifically, in the context of convergence towards equilibrium, by Corollary 17. Again, such inequalities may be useful in the study of weak turbulence [7].

**Proposition 16.** Let some $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $\varpi : \mathcal{E} \to \bar{J}$ and $r \in [1, 2]$. Then for any $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $f, g : \mathcal{E} \to \bar{J}$ such that all terms below are finite, we have

$$
(112) \quad \|(f-g) \varpi\|_{L^r(g \in J)}^2 \le \left( \int_0^1 (1-\tau) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right)^{-1} \mathcal{H}_{\Phi}[f|g],
$$

where $\mathcal{H}_{\Phi}$ is the $\Phi$-relative-entropy defined in (110).

*Proof.* Let us recall the definition of $\mathcal{H}_{\Phi}[f|g]$, that is

$$
\mathcal{H}_{\Phi}[f|g] = \int_{0}^{1} (1-\tau) \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g + \tau f) \, d\mu(\zeta) \right) d\tau.
$$
We fix $\tau \in (0, 1)$. Let $p = \frac{2}{r} \in [1, 2]$ and $q = \frac{2}{2-r} \in [2, +\infty]$, so that $\frac{1}{p} + \frac{1}{q} = 1$. By Hölder's inequality, we have

$$ \int_{g \in J} |f-g|^{\frac{2}{p}} \varpi^{\frac{2}{p}} d\mu(\zeta) = \int_{g \in J} |f-g|^{\frac{2}{p}} \Phi''((1-\tau)g + \tau f)^{\frac{1}{p}} \left[ \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right]^{\frac{1}{p}} d\mu(\zeta) \\ \le \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g + \tau f) d\mu(\zeta) \right)^{\frac{1}{p}} \left\| \left[ \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right]^{\frac{1}{p}} \right\|_{L^q(g \in J)}^{1/p}. $$

Raising the above inequality to the power $p$, we obtain (113)

$$ \left( \int_{g \in J} |f-g|^{\frac{2}{p}} \varpi^{\frac{2}{p}} d\mu(\zeta) \right)^p \le \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g + \tau f) d\mu(\zeta) \right) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^q(g \in J)}^{1/p}, $$

where we used the fact that $\| \cdot \|_{L^q} = \| \cdot \|_{L^\frac{q}{p}}$. Since $\frac{2}{p} = r$ and $\frac{q}{p} = \frac{r}{2-r}$, (113) actually writes

$$ \|(f-g)\varpi\|_{L^r(g \in J)}^2 \le \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g + \tau f) d\mu(\zeta) \right) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}.$$

If $\varpi$ is zero $\mu$-almost everywhere on $\{g \in J\}$, then the proposition is trivial. Else, since $\Phi'' > 0$ on $\{g \in J\}$ and $(1-\tau)g + \tau f \in J$ on $\{g \in J\}$ for any $\tau \in (0, 1)$, we know that $\|\Phi''((1-\tau)g + \tau f) \varpi^2\|_{L^{\frac{r}{2-r}}(g \in J)} > 0$. Since we assumed that the quantity

$$ \left( \int_0^1 (1-\tau) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right)^{-1} $$

is finite, we also know that for almost every $\tau \in (0, 1)$ we have $\left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)} < \infty$. For these values of $\tau$, we then have

$$ \|(f-g)\varpi\|_{L^r(g \in J)}^2 \frac{1-\tau}{\left\| \frac{\varpi^2}{\Phi''((1-\tau)g+\tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}} \le (1-\tau) \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g+\tau f) d\mu(\zeta) \right). $$

Integrating in $\tau$ yields

$$ \|(f-g)\varpi\|_{L^r(g \in J)}^2 \left( \int_0^1 (1-\tau) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right) \le \mathcal{H}_{\Phi}[f|g]. $$

Since, again by hypothesis, the integral in $\tau$ is nonzero (its inverse is finite), we obtain (112). □

Remarking that $\{M_{\Phi}^{\mathcal{F}} \in J\} = \mathcal{E}$ and that for any $f \in \mathcal{F}$, $\mathcal{H}_{\Phi}[f|M_{\Phi}^{\mathcal{F}}] = H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}]$ (see Proposition 15), we straightforwardly obtain the following corollary.

**Corollary 17.** Let some $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $\varpi : \mathcal{E} \to \bar{J}$ and $r \in [1, 2]$. With the same notations as in Proposition 15, assuming $M_{\Phi}^{\mathcal{F}}$ exists, we have for any $f \in \mathcal{F}$ such that the integral term below is finite,

$$ (114) \qquad \|(f - M_{\Phi}^{\mathcal{F}}) \varpi\|_{L^r}^2 \le \left( \int_0^1 (1-\tau) \left\| \frac{\varpi^2}{\Phi''((1-\tau)M_{\Phi}^{\mathcal{F}} + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right)^{-1} H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}], $$

where $H_{\Phi}$ is defined in (108)–(109) and $M_{\Phi}^{\mathcal{F}}$ is the equilibrium associated to $\Phi$ and the set $\mathcal{F}$, defined in (111).

## APPENDIX C. TECHNICAL RESULTS

In this section, we consider $M_{\varepsilon}^f$, the Fermi-Dirac distribution associated to some $\varepsilon > 0$ and $0 \le f \in L_2^1(\mathbb{R}^3)$ such that $1 - \varepsilon f \ge 0$. The existence of $M_{\varepsilon}^f$ is provided by assuming $\gamma > \frac{2}{5}$ (see [22]), where we recall the notation

$$ (115) \qquad \gamma := \frac{T}{T_F(\rho, \varepsilon)}, $$
We fix $\tau \in (0, 1)$. Let $p = \frac{2}{r} \in [1, 2]$ and $q = \frac{2}{2-r} \in [2, +\infty]$, so that $\frac{1}{p} + \frac{1}{q} = 1$. By Hölder's inequality, we have

$$ \int_{g \in J} |f - g|^{\frac{2}{p}} \varpi^{\frac{2}{p}} d\mu(\zeta) = \int_{g \in J} |f - g|^{\frac{2}{p}} \Phi''((1-\tau)g + \tau f)^{\frac{1}{p}} \left[ \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right]^{\frac{1}{p}} d\mu(\zeta) \\ \le \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g + \tau f) d\mu(\zeta) \right)^{\frac{1}{p}} \left\| \left[ \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right]^{\frac{1}{p}} \right\|_{L^q(g \in J)}^{1/p} $$

Raising the above inequality to the power $p$, we obtain (113)

$$ \left( \int_{g \in J} |f - g|^{\frac{2}{p}} \varpi^{\frac{2}{p}} d\mu(\zeta) \right)^p \le \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g + \tau f) d\mu(\zeta) \right) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g + \tau f)} \right\|_{L^q(g \in J)}^{1/p}, $$

where we used the fact that $\| \cdot \|_{L^q} = \| \cdot \|_{L^\frac{q}{p}}$. Since $\frac{2}{p} = r$ and $\frac{q}{p} = \frac{r}{2-r}$, (113) actually writes

$$ \|(f - g) \varpi\|_{L^r(g \in J)}^2 \le \left( \int_{g \in J} (f - g)^2 \Phi''((1 - \tau)g + \tau f) d\mu(\zeta) \right) \left\| \frac{\varpi^2}{\Phi''((1 - \tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)} $$

If $\varpi$ is zero $\mu$-almost everywhere on $\{g \in J\}$, then the proposition is trivial. Else, since $\Phi'' > 0$ on $\{g \in J\}$ and $(1-\tau)g + \tau f \in J$ on $\{g \in J\}$ for any $\tau \in (0, 1)$, we know that $\|\Phi''((1-\tau)g + \tau f) \varpi^2\|_{L^{\frac{r}{2-r}}(g \in J)} > 0$. Since we assumed that the quantity

$$ \left( \int_0^1 (1-\tau) \left\| \frac{\varpi^2}{\Phi''((1-\tau)g+\tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right)^{-1} $$

is finite, we also know that for almost every $\tau \in (0, 1)$ we have $\left\| \frac{\varpi^2}{\Phi''((1-\tau)g+\tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)} < \infty$. For these values of $\tau$, we then have

$$ \|(f - g) \varpi\|_{L^r(g \in J)}^2 \frac{1 - \tau}{\left\| \frac{\varpi^2}{\Phi''((1-\tau)g+\tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}} \le (1-\tau) \left( \int_{g \in J} (f-g)^2 \Phi''((1-\tau)g+\tau f) d\mu(\zeta) \right). $$

Integrating in $\tau$ yields

$$ \|(f - g) \varpi\|_{L^r(g \in J)}^2 \left( \int_0^1 (1 - \tau) \left\| \frac{\varpi^2}{\Phi''((1 - \tau)g + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right) \le \mathcal{H}_{\Phi}[f|g]. $$

Since, again by hypothesis, the integral in $\tau$ is nonzero (its inverse is finite), we obtain (112). □

Remarking that $\{M_{\Phi}^{\mathcal{F}} \in J\} = \mathcal{E}$ and that for any $f \in \mathcal{F}$, $\mathcal{H}_{\Phi}[f|M_{\Phi}^{\mathcal{F}}] = H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}]$ (see Proposition 15), we straightforwardly obtain the following corollary.

**Corollary 17.** Let some $(\mathcal{A}, \text{Bor}(\bar{J}))$-measurable $\varpi : \mathcal{E} \to \bar{J}$ and $r \in [1, 2]$. With the same notations as in Proposition 15, assuming $M_{\Phi}^{\mathcal{F}}$ exists, we have for any $f \in \mathcal{F}$ such that the integral term below is finite,

$$ (114) \qquad \|(f - M_{\Phi}^{\mathcal{F}}) \varpi\|_{L^r}^2 \le \left( \int_0^1 (1-\tau) \left\| \frac{\varpi^2}{\Phi''((1-\tau)M_{\Phi}^{\mathcal{F}} + \tau f)} \right\|_{L^{\frac{r}{2-r}}(g \in J)}^{-1} d\tau \right)^{-1} H_{\Phi}[f|M_{\Phi}^{\mathcal{F}}], $$

where $H_{\Phi}$ is defined in (108)–(109) and $M_{\Phi}^{\mathcal{F}}$ is the equilibrium associated to $\Phi$ and the set $\mathcal{F}$, defined in (111).

## APPENDIX C. TECHNICAL RESULTS

In this section, we consider $M_{\varepsilon}^f$, the Fermi-Dirac distribution associated to some $\varepsilon > 0$ and $0 \le f \in L_2^1(\mathbb{R}^3)$ such that $1 - \varepsilon f \ge 0$. The existence of $M_{\varepsilon}^f$ is provided by assuming $\gamma > \frac{2}{5}$ (see [22]), where we recall the notation

$$ (115) \qquad \gamma := \frac{T}{T_F(\rho, \varepsilon)}, $$
where $T_F(\rho, \varepsilon) = \frac{1}{2} \left(\frac{3\rho\varepsilon}{4\pi}\right)^{2/3}$ is the Fermi temperature associated to $\rho$ and $\varepsilon$; and $\rho, T$ are respectively the density and temperature associated to the distribution $f$, defined in (12). We also recall the notation, for $x \in [0, \varepsilon^{-1})$,

$$ \varphi_{\varepsilon}(x) = \frac{x}{1 - \varepsilon x}. $$

**C.1.** $L^{\infty}$ bound for the Fermi-Dirac statistics. In this subsection, we provide an $L^{\infty}$ bound on the Fermi-Dirac statistics. The following result is very similar to [3, Lemma A.1].

**Proposition 18.** Let $\varepsilon > 0$ and $f \in L_2^1(\mathbb{R}^3)$ be a nonnegative distribution such that $1 - \varepsilon f \ge 0$ and $\gamma \ge \gamma^{\dagger}$, where $\gamma$ is given by (115) and

$$ (116) \qquad \gamma^{\dagger} := \left(\frac{4}{\pi}\right)^{\frac{1}{3}} \left(\frac{5}{3}\right)^{\frac{5}{3}}. $$

Then the quantity $\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}$ satisfies

$$ (117) \qquad \varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty} \le \frac{2}{3} \left(\frac{\gamma}{\gamma^{\dagger}}\right)^{-3/2}. $$

*Proof.* Our proof is based on [22, proof of Proposition 3]. We introduce, for $s \ge 0$,

$$ I_s(t) := \int_0^\infty \frac{r^s}{1 + te^{r^2}} \, dr, \quad P(t) := I_4(t)[I_2(t)]^{-5/3}, \quad t > 0. $$

It is proven in [22, proof of Proposition 3] that $P$ is continuous and increasing on $\mathbb{R}_+$, and that

$$ P\left(\frac{1}{\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}}\right) = 3\rho^{-2/3} T (4\pi\varepsilon^{-1})^{2/3} \equiv \frac{3^{5/3}}{2} \gamma. $$

Let $\alpha > 0$ and

$$ t = \alpha^{-1} \times \frac{3\sqrt{\pi}}{4} \gamma^{3/2}. $$

As, for any $r \ge 0$, it holds that

$$ \frac{e^{-r^2}}{1 + t} \le \frac{1}{1 + t e^{r^2}} \le \frac{e^{-r^2}}{t}, $$

we have

$$ P(t) \le \left(\frac{1}{t} \int_0^\infty r^4 e^{-r^2} \, dr\right) \left(\frac{1}{1+t} \int_0^\infty r^2 e^{-r^2} \, dr\right)^{-5/3} = \frac{(1+t)^{5/3}}{t} \left(\frac{1}{2}\Gamma\left(\frac{5}{2}\right)\right) \left(\frac{1}{2}\Gamma\left(\frac{3}{2}\right)\right)^{-5/3}, $$

that is

$$ P(t) \le \frac{(1+t)^{5/3}}{t} \times \frac{3 \times 2^{1/3}}{\pi^{1/3}}. $$

We define

$$ \gamma^{\alpha} := \left( \frac{4}{3\sqrt{\pi}} \times \frac{\alpha}{\alpha^{2/5} - 1} \right)^{2/3}. $$

Then, whenever $\gamma \ge \gamma^{\alpha}$, we have $t \ge \frac{1}{\alpha^{2/5} - 1}$, so that $1 + t \le \alpha^{2/5} t$, implying

$$ P(t) \le t^{2/3} \times \frac{3 \times 2^{1/3} \alpha^{2/3}}{\pi^{1/3}} = \alpha^{-2/3} \times \left(\frac{3\sqrt{\pi}}{4}\right)^{2/3} \gamma \times \frac{3 \times 2^{1/3} \alpha^{2/3}}{\pi^{1/3}} = \frac{3^{5/3}}{2} \gamma = P\left(\frac{1}{\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}}\right). $$

Since $P$ is increasing, we deduce that, whenever $\gamma \ge \gamma^{\alpha}$, we have

$$ \alpha^{-1} \times \frac{3\sqrt{\pi}}{4} \gamma^{3/2} = t \le \frac{1}{\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}}, $$

that is

$$ (118) \qquad \varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty} \le \alpha \times \frac{4}{3\sqrt{\pi}} \gamma^{-3/2}. $$
where $T_F(\rho, \varepsilon) = \frac{1}{2} \left(\frac{3\rho\varepsilon}{4\pi}\right)^{2/3}$ is the Fermi temperature associated to $\rho$ and $\varepsilon$; and $\rho, T$ are respectively the density and temperature associated to the distribution $f$, defined in (12). We also recall the notation, for $x \in [0, \varepsilon^{-1})$,

$$ \varphi_{\varepsilon}(x) = \frac{x}{1 - \varepsilon x}. $$

**C.1.** $L^{\infty}$ bound for the Fermi-Dirac statistics. In this subsection, we provide an $L^{\infty}$ bound on the Fermi-Dirac statistics. The following result is very similar to [3, Lemma A.1].

**Proposition 18.** Let $\varepsilon > 0$ and $f \in L_2^1(\mathbb{R}^3)$ be a nonnegative distribution such that $1 - \varepsilon f \ge 0$ and $\gamma \ge \gamma^{\dagger}$, where $\gamma$ is given by (115) and

$$ (116) \qquad \gamma^{\dagger} := \left(\frac{4}{\pi}\right)^{\frac{1}{3}} \left(\frac{5}{3}\right)^{\frac{5}{3}}. $$

Then the quantity $\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}$ satisfies

$$ (117) \qquad \varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty} \le \frac{2}{3} \left(\frac{\gamma}{\gamma^{\dagger}}\right)^{-3/2}. $$

*Proof.* Our proof is based on [22, proof of Proposition 3]. We introduce, for $s \ge 0$,

$$ I_s(t) := \int_0^\infty \frac{r^s}{1 + te^{r^2}} \, dr, \quad P(t) := I_4(t)[I_2(t)]^{-5/3}, \quad t > 0. $$

It is proven in [22, proof of Proposition 3] that $P$ is continuous and increasing on $\mathbb{R}_+$, and that

$$ P\left(\frac{1}{\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}}\right) = 3\rho^{-2/3} T (4\pi\varepsilon^{-1})^{2/3} = \frac{3^{5/3}}{2} \gamma. $$

Let $\alpha > 0$ and

$$ t = \alpha^{-1} \times \frac{3\sqrt{\pi}}{4} \gamma^{3/2}. $$

As, for any $r \ge 0$, it holds that

$$ \frac{e^{-r^2}}{1 + t} \le \frac{1}{1 + t e^{r^2}} \le \frac{e^{-r^2}}{t}, $$

we have

$$ P(t) \le \left(\frac{1}{t} \int_0^\infty r^4 e^{-r^2} \, dr\right) \left(\frac{1}{1+t} \int_0^\infty r^2 e^{-r^2} \, dr\right)^{-5/3} = \frac{(1+t)^{5/3}}{t} \left(\frac{1}{2}\Gamma\left(\frac{5}{2}\right)\right) \left(\frac{1}{2}\Gamma\left(\frac{3}{2}\right)\right)^{-5/3}, $$

that is

$$ P(t) \le \frac{(1+t)^{5/3}}{t} \times \frac{3 \times 2^{1/3}}{\pi^{1/3}}. $$

We define

$$ \gamma^{\alpha} := \left( \frac{4}{3\sqrt{\pi}} \times \frac{\alpha}{\alpha^{2/5} - 1} \right)^{2/3}. $$

Then, whenever $\gamma \ge \gamma^{\alpha}$, we have $t \ge \frac{1}{\alpha^{2/5} - 1}$, so that $1 + t \le \alpha^{2/5} t$, implying

$$ P(t) \le t^{2/3} \times \frac{3 \times 2^{1/3} \alpha^{2/3}}{\pi^{1/3}} = \alpha^{-2/3} \times \left(\frac{3\sqrt{\pi}}{4}\right)^{2/3} \gamma \times \frac{3 \times 2^{1/3} \alpha^{2/3}}{\pi^{1/3}} = \frac{3^{5/3}}{2} \gamma = P\left(\frac{1}{\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}}\right). $$

Since $P$ is increasing, we deduce that, whenever $\gamma \ge \gamma^{\alpha}$, we have

$$ \alpha^{-1} \times \frac{3\sqrt{\pi}}{4} \gamma^{3/2} = t \le \frac{1}{\varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty}}, $$

that is

$$ (118) \qquad \varepsilon \|\varphi_{\varepsilon}(\mathcal{M}_{\varepsilon}^{f})\|_{\infty} \le \alpha \times \frac{4}{3\sqrt{\pi}} \gamma^{-3/2}. $$
By computing the derivative of $\alpha \mapsto \frac{\alpha}{\alpha^{2/5}-1}$, we can minimize $\alpha \mapsto \gamma^\alpha$ and find that the minimum value is

$$ \gamma^\dagger = \left(\frac{4}{\pi}\right)^{\frac{1}{3}} \left(\frac{5}{3}\right)^{\frac{5}{3}}, $$

reached for $\alpha^\dagger = (\frac{5}{3})^{5/2}$, and, combined with (118), this proves (117). □

**C.2. Regularity in $\varepsilon$ of the coefficients of the Fermi-Dirac statistics.** In this last subsection, for any $\varepsilon \ge 0$ and $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$, we denote by $M_\varepsilon \equiv M_\varepsilon^{\varphi_\varepsilon^{-1}(g)}$ the $\varepsilon$-Fermi distribution associated to $\varphi_\varepsilon^{-1}(g)$. In particular, in the limit case $\varepsilon = 0$, $M_0$ is the Maxwellian distribution associated to $g$. We also denote $a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon$ and $\rho_\varepsilon, u_\varepsilon, T_\varepsilon$ the quantities such that, letting

$$ (119) \qquad M_\varepsilon \equiv M_\varepsilon^{\frac{g}{1+\varepsilon g}}, \qquad M_\varepsilon = \frac{M_\varepsilon}{1-\varepsilon M_\varepsilon}, $$

we have, for any $v \in \mathbb{R}^3$,

$$ (120) \qquad M_\varepsilon(v) = \exp(a_\varepsilon + b_\varepsilon |v - \bar{u}_\varepsilon|^2), $$

and

$$ (121) \qquad \int_{\mathbb{R}^3} \frac{g}{1+\varepsilon g} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho_\varepsilon \\ \rho_\varepsilon u_\varepsilon \\ 3\rho_\varepsilon T_\varepsilon + \rho_\varepsilon |u_\varepsilon|^2 \end{pmatrix}. $$

**Lemma 19.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notation (121), the application $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ is continuous on $\mathbb{R}_+$ and $C^1$ on $\mathbb{R}_+^*$.

*Proof.* The continuity of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ on $\mathbb{R}_+$ comes by dominated convergence, as it holds for any $\varepsilon \ge 0$ and $v \in \mathbb{R}^3$ that

$$ \frac{g}{1+\varepsilon g} (1+|v|^2) \le g(1+|v|^2), $$

and by hypothesis $0 \le g \in L_2^1(\mathbb{R}^3)$. Similarly, for any $\varepsilon > 0$ and $v \in \mathbb{R}^3$ we have

$$ \left| \partial_{\varepsilon} \frac{g}{1+\varepsilon g} \right| (1+|v|^2) = \left( \frac{g}{1+\varepsilon g} \right)^2 (1+|v|^2) \le \varepsilon^{-1} g (1+|v|^2). $$

Therefore, for any $\varepsilon > 0$, we have

$$ \sup_{\varepsilon_* \in (\varepsilon/2, 3\varepsilon/2)} \left| \partial_{\varepsilon_*} \frac{g}{1+\varepsilon_* g} \right| (1+|v|^2) \le 2\varepsilon^{-1} g (1+|v|^2). $$

The differentiability of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ on $\mathbb{R}_+^*$ then comes by dominated convergence, as $g \in L_2^1(\mathbb{R}^3)$. □

The following lemmas provide the continuity of $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon)$ at the point $\varepsilon = 0$ and its differentiability on $\mathbb{R}_+^*$.

**Lemma 20.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notations (119)-(120), the application $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon)$ is continuous at the point $\varepsilon = 0$.

*Proof.* Using the notations (119) and (121), we have

$$ (122) \qquad \left| \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv - \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \right| = \left| \int_{\mathbb{R}^3} \frac{\varepsilon M_\varepsilon^2}{1+\varepsilon M_\varepsilon} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \right| \le \varepsilon \|M_\varepsilon\|_\infty \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ |v| \\ |v|^2 \end{pmatrix} dv \\ \le \varepsilon \|M_\varepsilon\|_\infty (\rho_\varepsilon + 3\rho_\varepsilon T_\varepsilon + \rho_\varepsilon |u_\varepsilon|^2). $$

Recall the notation, in this case,

$$ \gamma_\varepsilon = \frac{T_\varepsilon}{T_F(\rho_\varepsilon, \varepsilon)}, $$
By computing the derivative of $\alpha \mapsto \frac{\alpha}{\alpha^{2/5} - 1}$, we can minimize $\alpha \mapsto \gamma^\alpha$ and find that the minimum value is

$$ \gamma^\dagger = \left(\frac{4}{\pi}\right)^{\frac{1}{3}} \left(\frac{5}{3}\right)^{\frac{5}{3}}, $$

reached for $\alpha^\dagger = (\frac{5}{3})^{5/2}$, and, combined with (118), this proves (117). □

**C.2. Regularity in $\varepsilon$ of the coefficients of the Fermi-Dirac statistics.** In this last subsection, for any $\varepsilon \ge 0$ and $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$, we denote by $M_\varepsilon \equiv M_\varepsilon^{\varphi_\varepsilon^{-1}(g)}$ the $\varepsilon$-Fermi distribution associated to $\varphi_\varepsilon^{-1}(g)$. In particular, in the limit case $\varepsilon = 0$, $M_0$ is the Maxwellian distribution associated to $g$. We also denote $a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon$ and $\rho_\varepsilon, u_\varepsilon, T_\varepsilon$ the quantities such that, letting

$$ (119) \qquad M_\varepsilon \equiv M_\varepsilon^{\frac{g}{1+\varepsilon g}}, \qquad M_\varepsilon = \frac{M_\varepsilon}{1 - \varepsilon M_\varepsilon}, $$

we have, for any $v \in \mathbb{R}^3$,

$$ (120) \qquad M_\varepsilon(v) = \exp(a_\varepsilon + b_\varepsilon |v - \bar{u}_\varepsilon|^2), $$

and

$$ (121) \qquad \int_{\mathbb{R}^3} \frac{g}{1 + \varepsilon g} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho_\varepsilon \\ \rho_\varepsilon u_\varepsilon \\ 3\rho_\varepsilon T_\varepsilon + \rho_\varepsilon |u_\varepsilon|^2 \end{pmatrix}. $$

**Lemma 19.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notation (121), the application $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ is continuous on $\mathbb{R}_+$ and $C^1$ on $\mathbb{R}_+^*$.

*Proof.* The continuity of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ on $\mathbb{R}_+$ comes by dominated convergence, as it holds for any $\varepsilon \ge 0$ and $v \in \mathbb{R}^3$ that

$$ \frac{g}{1 + \varepsilon g} (1 + |v|^2) \le g (1 + |v|^2), $$

and by hypothesis $0 \le g \in L_2^1(\mathbb{R}^3)$. Similarly, for any $\varepsilon > 0$ and $v \in \mathbb{R}^3$ we have

$$ \left| \partial_{\varepsilon} \frac{g}{1 + \varepsilon g} \right| (1 + |v|^2) = \left( \frac{g}{1 + \varepsilon g} \right)^2 (1 + |v|^2) \le \varepsilon^{-1} g (1 + |v|^2). $$

Therefore, for any $\varepsilon > 0$, we have

$$ \sup_{\varepsilon_* \in (\varepsilon/2, 3\varepsilon/2)} \left| \partial_{\varepsilon_*} \frac{g}{1 + \varepsilon_* g} \right| (1 + |v|^2) \le 2\varepsilon^{-1} g (1 + |v|^2). $$

The differentiability of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ on $\mathbb{R}_+^*$ then comes by dominated convergence, as $g \in L_2^1(\mathbb{R}^3)$. □

The following lemmas provide the continuity of $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon)$ at the point $\varepsilon = 0$ and its differentiability on $\mathbb{R}_+^*$.

**Lemma 20.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notations (119)-(120), the application $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon)$ is continuous at the point $\varepsilon = 0$.

*Proof.* Using the notations (119) and (121), we have

$$ (122) \qquad \left| \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv - \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \right| = \left| \int_{\mathbb{R}^3} \frac{\varepsilon M_\varepsilon^2}{1 + \varepsilon M_\varepsilon} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \right| \le \varepsilon \|M_\varepsilon\|_\infty \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ |v| \\ |v|^2 \end{pmatrix} dv \\ \le \varepsilon \|M_\varepsilon\|_\infty (\rho_\varepsilon + 3\rho_\varepsilon T_\varepsilon + \rho_\varepsilon |u_\varepsilon|^2). $$

Recall the notation, in this case,

$$ \gamma_\varepsilon = \frac{T_\varepsilon}{T_F(\rho_\varepsilon, \varepsilon)}, $$
where $T_F(\rho_\varepsilon, \varepsilon) = \frac{1}{2} \left( \frac{3\rho_\varepsilon \varepsilon}{4\pi} \right)^{2/3}$. By continuity at the point $\varepsilon = 0$ of the application $\varepsilon \mapsto (\rho_\varepsilon, T_\varepsilon)$, given by Lemma 19, we have

$$ \gamma_{\varepsilon} \xrightarrow{\varepsilon \to 0} +\infty. $$

Thereby, there exists $\varepsilon^* > 0$ such that $\gamma_\varepsilon \ge \gamma^\dagger$ for any $\varepsilon \in (0, \varepsilon^*)$, where $\gamma^\dagger$ is a universal constant defined in Proposition 18. Then, from (117) in Proposition 18, for any $\varepsilon \in (0, \varepsilon^*)$,

$$ \varepsilon \| M_\varepsilon \|_\infty \le \frac{2}{3} \left( \frac{\gamma_\varepsilon}{\gamma_*} \right)^{-\frac{3}{2}}, $$

which vanishes as $\varepsilon \to 0$, since $\gamma_\varepsilon$ tends to $+\infty$ in this limit. Combining this result, the continuity of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ at $\varepsilon = 0$, given by Lemma 19, and Equation (122), we obtain

$$ \left| \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv - \int_{\mathbb{R}^3} \mathcal{M}_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \right| \xrightarrow{\varepsilon \to 0} 0. $$

The continuity of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ at $\varepsilon = 0$ being equivalent to the statement

$$ \int_{\mathbb{R}^3} \mathcal{M}_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \xrightarrow{\varepsilon \to 0} \int_{\mathbb{R}^3} \mathcal{M}_0 \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv, $$

we finally conclude that

$$ \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \xrightarrow{\varepsilon \to 0} \int_{\mathbb{R}^3} \mathcal{M}_0 \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv. $$

Both $M_\varepsilon$ and $\mathcal{M}_0$ are gaussian distributions, which coefficients are continuously defined by the above moments, allowing to conclude to the continuity of these coefficients at the point $\varepsilon = 0$. $\square$

**Lemma 21.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notations (119)-(120), the application $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon)$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$.

*Proof.* Since the distribution $M_\varepsilon(\cdot + \bar{u}_\varepsilon)$ is radially symmetric, so is $\mathcal{M}_\varepsilon(\cdot + \bar{u}_\varepsilon)$, hence $\bar{u}_\varepsilon = u_\varepsilon$, which, by Lemma 19, is $\mathcal{C}^1$ on $\mathbb{R}_+^*$. We then define

$$ g_\varepsilon := g(\cdot + u_\varepsilon), \quad \mathcal{N}_\varepsilon := \mathcal{M}_\varepsilon(\cdot + u_\varepsilon) \equiv \mathcal{M}_\varepsilon^{\varphi_\varepsilon^{-1}(g_\varepsilon)} \quad \text{and} \quad N_\varepsilon := M_\varepsilon(\cdot + u_\varepsilon), $$

so that, for any $\varepsilon > 0$, it holds that

$$ N_\varepsilon = \exp(a_\varepsilon + b_\varepsilon |v|^2). $$

As in (121), we let $\rho_\varepsilon, T_\varepsilon > 0$ be such that

$$ \int_{\mathbb{R}^3} \frac{g_\varepsilon}{1 + \varepsilon g_\varepsilon} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho_\varepsilon \\ 0 \\ 3\rho_\varepsilon T_\varepsilon \end{pmatrix}. $$

Let us now show that $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon)$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$. It is proven in [22, proof of Proposition 3] that, letting

$$ I_s(\tau) := \int_0^\infty \frac{r^s}{1 + \tau e^{r^2}} dr, \quad P(\tau) := I_4(\tau) I_2(\tau)^{-5/3}, \quad \tau \in \mathbb{R}_+^*, $$

the function $P$ is an increasing $\mathcal{C}^1$ function from $\mathbb{R}_+^*$ to $(\frac{3^{5/3}}{5}, +\infty)$, with $P' > 0$ on $\mathbb{R}_+^*$. Therefore it is invertible, and $P^{-1}$ is also $\mathcal{C}^1$. All the more, a dominated convergence argument ensures that $I_2$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$. It is moreover shown in [22, proof of Proposition 3] that

$$ \left(\frac{\varepsilon}{4\pi}\right)^{2/3} P\left(\frac{1}{\varepsilon e^{a_\varepsilon}}\right) = \frac{3\rho_\varepsilon T_\varepsilon}{\rho_\varepsilon^{5/3}}, \quad b_\varepsilon = -\left(\frac{4\pi}{\varepsilon\rho_\varepsilon} I_2\left(\frac{1}{\varepsilon e^{a_\varepsilon}}\right)\right)^{\frac{2}{3}}. $$

By Lemma 19, the application $\varepsilon \mapsto (\rho_\varepsilon, T_\varepsilon)$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$, hence so is the application $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon)$, as a composition of $\mathcal{C}^1$ applications. $\square$
where $T_F(\rho_\varepsilon, \varepsilon) = \frac{1}{2} \left( \frac{3\rho_\varepsilon \varepsilon}{4\pi} \right)^{2/3}$. By continuity at the point $\varepsilon = 0$ of the application $\varepsilon \mapsto (\rho_\varepsilon, T_\varepsilon)$, given by Lemma 19, we have

$$ \gamma_{\varepsilon} \xrightarrow{\varepsilon \to 0} +\infty. $$

Thereby, there exists $\varepsilon^* > 0$ such that $\gamma_\varepsilon \ge \gamma^\dagger$ for any $\varepsilon \in (0, \varepsilon^*)$, where $\gamma^\dagger$ is a universal constant defined in Proposition 18. Then, from (117) in Proposition 18, for any $\varepsilon \in (0, \varepsilon^*)$,

$$ \varepsilon \| M_\varepsilon \|_{\infty} \le \frac{2}{3} \left( \frac{\gamma_\varepsilon}{\gamma_*} \right)^{-\frac{3}{2}}, $$

which vanishes as $\varepsilon \to 0$, since $\gamma_\varepsilon$ tends to $+\infty$ in this limit. Combining this result, the continuity of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ at $\varepsilon = 0$, given by Lemma 19, and Equation (122), we obtain

$$ \left| \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv - \int_{\mathbb{R}^3} \mathcal{M}_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \right| \xrightarrow{\varepsilon \to 0} 0. $$

The continuity of $\varepsilon \mapsto (\rho_\varepsilon, u_\varepsilon, T_\varepsilon)$ at $\varepsilon = 0$ being equivalent to the statement

$$ \int_{\mathbb{R}^3} \mathcal{M}_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \xrightarrow{\varepsilon \to 0} \int_{\mathbb{R}^3} \mathcal{M}_0 \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv, $$

we finally conclude that

$$ \int_{\mathbb{R}^3} M_\varepsilon \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv \xrightarrow{\varepsilon \to 0} \int_{\mathbb{R}^3} \mathcal{M}_0 \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv. $$

Both $M_\varepsilon$ and $\mathcal{M}_0$ are gaussian distributions, which coefficients are continuously defined by the above moments, allowing to conclude to the continuity of these coefficients at the point $\varepsilon = 0$. $\square$

**Lemma 21.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notations (119)-(120), the application $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon, \bar{u}_\varepsilon)$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$.

*Proof.* Since the distribution $M_\varepsilon(\cdot + \bar{u}_\varepsilon)$ is radially symmetric, so is $\mathcal{M}_\varepsilon(\cdot + \bar{u}_\varepsilon)$, hence $\bar{u}_\varepsilon = u_\varepsilon$, which, by Lemma 19, is $\mathcal{C}^1$ on $\mathbb{R}_+^*$. We then define

$$ g_\varepsilon := g(\cdot + u_\varepsilon), \quad \mathcal{N}_\varepsilon := \mathcal{M}_\varepsilon(\cdot + u_\varepsilon) \equiv \mathcal{M}_\varepsilon^{\varphi_\varepsilon^{-1}(g_\varepsilon)} \quad \text{and} \quad N_\varepsilon := M_\varepsilon(\cdot + u_\varepsilon), $$

so that, for any $\varepsilon > 0$, it holds that

$$ N_\varepsilon = \exp(a_\varepsilon + b_\varepsilon |v|^2). $$

As in (121), we let $\rho_\varepsilon, T_\varepsilon > 0$ be such that

$$ \int_{\mathbb{R}^3} \frac{g_\varepsilon}{1 + \varepsilon g_\varepsilon} \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} dv = \begin{pmatrix} \rho_\varepsilon \\ 0 \\ 3\rho_\varepsilon T_\varepsilon \end{pmatrix}. $$

Let us now show that $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon)$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$. It is proven in [22, proof of Proposition 3] that, letting

$$ I_s(\tau) := \int_0^\infty \frac{r^s}{1 + \tau e^{r^2}} dr, \quad P(\tau) := I_4(\tau) I_2(\tau)^{-5/3}, \quad \tau \in \mathbb{R}_+^*, $$

the function $P$ is an increasing $\mathcal{C}^1$ function from $\mathbb{R}_+^*$ to $(\frac{3^{5/3}}{5}, +\infty)$, with $P' > 0$ on $\mathbb{R}_+^*$. Therefore it is invertible, and $P^{-1}$ is also $\mathcal{C}^1$. All the more, a dominated convergence argument ensures that $I_2$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$. It is moreover shown in [22, proof of Proposition 3] that

$$ \left(\frac{\varepsilon}{4\pi}\right)^{2/3} P\left(\frac{1}{\varepsilon e^{a_\varepsilon}}\right) = \frac{3\rho_\varepsilon T_\varepsilon}{\rho_\varepsilon^{5/3}}, \quad b_\varepsilon = -\left(\frac{4\pi}{\varepsilon\rho_\varepsilon} I_2\left(\frac{1}{\varepsilon e^{a_\varepsilon}}\right)\right)^{\frac{2}{3}}. $$

By Lemma 19, the application $\varepsilon \mapsto (\rho_\varepsilon, T_\varepsilon)$ is $\mathcal{C}^1$ on $\mathbb{R}_+^*$, hence so is the application $\varepsilon \mapsto (a_\varepsilon, b_\varepsilon)$, as a composition of $\mathcal{C}^1$ applications. $\square$
**Lemma 22.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notation (119), for any $\bar{\epsilon} > 0$, there exist $C > 0$ and $\eta > 0$ such that for any $\epsilon \in [0, \bar{\epsilon}]$ and $v \in \mathbb{R}^3$, we have

$$ (123) \qquad M_{\epsilon}(v) \le C e^{-\eta|v|^2}, $$

and

$$ (124) \qquad |\log M_{\epsilon}(v)| \le C(1 + |v|^2). $$

*Proof.* We denote

$$ \underline{a} = \sup_{\epsilon \in [0, \bar{\epsilon}]} |a_{\epsilon}|, \quad \underline{b} = - \sup_{\epsilon \in [0, \bar{\epsilon}]} |b_{\epsilon}|, \quad \bar{b} = - \inf_{\epsilon \in [0, \bar{\epsilon}]} |b_{\epsilon}| \quad \text{and} \quad \underline{u} = \sup_{\epsilon \in [0, \bar{\epsilon}]} |u_{\epsilon}|. $$

Combining the results of Lemmas 20 and 21, the application $\epsilon \mapsto (a_{\epsilon}, b_{\epsilon}, u_{\epsilon})$ is continuous on $\mathbb{R}_+$, from which we deduce that $\underline{a}, \underline{b}$ and $\underline{u}$ are finite.

Moreover, as $M_{\epsilon} \in L^1(\mathbb{R}^3)$ for all $\epsilon \in [0, \bar{\epsilon}]$, the application $\epsilon \mapsto b_{\epsilon}$ is (strictly) negative on $[0, \bar{\epsilon}]$, so that, as $\epsilon \mapsto b_{\epsilon}$ is continuous on $[0, \bar{\epsilon}]$, we have $\bar{b} < 0$.

Therefore, for any $0 \le \epsilon \le \bar{\epsilon}$ and $v \in \mathbb{R}^3$, we have

$$ |\log M_{\epsilon}(v)| = |a_{\epsilon} + b_{\epsilon}|v - u_{\epsilon}|^2 \le \underline{a} + 2|\underline{b}|\underline{u}^2 + 2|\underline{b}|v|^2, $$

and, since $|v - u_{\epsilon}|^2 \ge \frac{1}{2}|v|^2 - |u_{\epsilon}|^2 \ge \frac{1}{2}|v|^2 - \underline{u}^2$ and $\bar{b} < 0$,

$$ M_{\epsilon}(v) = e^{a_{\epsilon}+b_{\epsilon}|v-u_{\epsilon}|^2} \le e^{a+\bar{b}|v-u_{\epsilon}|^2} \le e^{a+|\bar{b}|\underline{u}^2+\frac{1}{2}\bar{b}|v|^2}. $$

Letting $\eta = -\frac{1}{2}\bar{b}$ and $C = \max(\underline{a} + 2|\underline{b}|\underline{u}^2, e^{a+|\bar{b}|\underline{u}^2})$ yields the result. $\square$

<figure><img src="image_1.png" /></figure>

## ACKNOWLEDGEMENTS

I thank Bertrand Lods with whom I had fruitful discussions and Matthieu Dolbeault who suggested to me the proof of the nice inequality (92).

I have no competing interests to declare that are relevant to the content of this article.

## REFERENCES

[1] R. Alonso, V. Bagland, L. Desvillettes, and B. Lods. About the use of entropy production for the Landau-Fermi-Dirac equation. *J. Stat. Phys.*, 183(1):Paper No. 10, 27, 2021.

[2] R. Alonso, V. Bagland, L. Desvillettes, and B. Lods. About the Landau-Fermi-Dirac equation with moderately soft potentials. *Arch. Ration. Mech. Anal.*, 244(3):779-875, 2022.

[3] R. Alonso, V. Bagland, and B. Lods. Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials. *J. Differential Equations*, 270:596-663, 2021.

[4] R. Alonso, V. Bagland, and B. Lods. Long time dynamics for the landau-fermi-dirac equation with hard potentials. *Journal of Differential Equations*, 270:596-663, 2021.

[5] A. V. Bobylev and C. Cercignani. On the rate of entropy production for the Boltzmann equation. *J. Statist. Phys.*, 94(3-4):603-618, 1999.

[6] F. Bolley and C. Villani. Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. *Ann. Fac. Sci. Toulouse Math. (6)*, 14(3):331-352, 2005.

[7] M. Breden and L. Desvillettes. Rigorous study of the equilibria of collision kernels appearing in the theory of weak turbulence. *Arch. Ration. Mech. Anal.*, 235(2):1151-1176, 2020.

[8] M. J. Cáceres, J. A. Carrillo, and J. Dolbeault. Nonlinear stability in $L^p$ for a confined system of charged particles. *SIAM J. Math. Anal.*, 34(2):478-494, 2002.

[9] María J. Cáceres, José A. Carrillo, and Jean Dolbeault. A generalization of the Csiszár-Kullback inequality. https://www.ceremade.dauphine.fr/~dolbeaul/Teaching/files/SNS-2012/Csiszar-Kullback.pdf.

[10] K. Carrapatoso, L. Desvillettes, and L. He. Estimates for the large time behavior of the Landau equation in the Coulomb case. *Arch. Ration. Mech. Anal.*, 224(2):381-420, 2017.

[11] C. Cercignani. H-theorem and trend to equilibrium in the kinetic theory of gases. *Arch. Mech. (Arch. Mech. Stos.)*, 34(3):231-241 (1983), 1982.

[12] S. Chapman and T. G. Cowling. *The Mathematical Theory of Non-uniform Gases*. Cambridge University Press, Cambridge, 1939.

[13] L. Desvillettes. Entropy dissipation estimates for the Landau equation in the Coulomb case and applications. *J. Funct. Anal.*, 269(5):1359-1403, 2015.

[14] L. Desvillettes. Structure entropique du noyau de collision de Landau. In *Séminaire Laurent Schwartz-Équations aux dérivées partielles et applications. Année 2014-2015*, pages Exp. No. XIV, 9. Ed. Éc. Polytech., Palaiseau, 2016.
**Lemma 22.** Let $0 \le g \in L_2^1(\mathbb{R}^3) \cap L \log L(\mathbb{R}^3)$. Using the notation (119), for any $\bar{\epsilon} > 0$, there exist $C > 0$ and $\eta > 0$ such that for any $\epsilon \in [0, \bar{\epsilon}]$ and $v \in \mathbb{R}^3$, we have

$$ (123) \qquad M_{\epsilon}(v) \le C e^{-\eta|v|^2}, $$

and

$$ (124) \qquad |\log M_{\epsilon}(v)| \le C(1 + |v|^2). $$

*Proof.* We denote

$$ \underline{a} = \sup_{\epsilon \in [0, \bar{\epsilon}]} |a_{\epsilon}|, \quad \underline{b} = - \sup_{\epsilon \in [0, \bar{\epsilon}]} |b_{\epsilon}|, \quad \bar{b} = - \inf_{\epsilon \in [0, \bar{\epsilon}]} |b_{\epsilon}| \quad \text{and} \quad \underline{u} = \sup_{\epsilon \in [0, \bar{\epsilon}]} |u_{\epsilon}|. $$

Combining the results of Lemmas 20 and 21, the application $\epsilon \mapsto (a_{\epsilon}, b_{\epsilon}, u_{\epsilon})$ is continuous on $\mathbb{R}_+$, from which we deduce that $\underline{a}$, $\underline{b}$ and $\underline{u}$ are finite.

Moreover, as $M_{\epsilon} \in L^1(\mathbb{R}^3)$ for all $\epsilon \in [0, \bar{\epsilon}]$, the application $\epsilon \mapsto b_{\epsilon}$ is (strictly) negative on $[0, \bar{\epsilon}]$, so that, as $\epsilon \mapsto b_{\epsilon}$ is continuous on $[0, \bar{\epsilon}]$, we have $\bar{b} < 0$.

Therefore, for any $0 \le \epsilon \le \bar{\epsilon}$ and $v \in \mathbb{R}^3$, we have

$$ |\log M_{\epsilon}(v)| = |a_{\epsilon} + b_{\epsilon}|v - u_{\epsilon}|^2 \le \underline{a} + 2|\underline{b}|\underline{u}^2 + 2|\underline{b}|v|^2, $$

and, since $|v - u_{\epsilon}|^2 \ge \frac{1}{2}|v|^2 - |u_{\epsilon}|^2 \ge \frac{1}{2}|v|^2 - \underline{u}^2$ and $\bar{b} < 0$,

$$ M_{\epsilon}(v) = e^{a_{\epsilon}+b_{\epsilon}|v-u_{\epsilon}|^2} \le e^{a+\bar{b}|v-u_{\epsilon}|^2} \le e^{a+|\bar{b}|\underline{u}^2+\frac{1}{2}\bar{b}|v|^2}. $$

Letting $\eta = -\frac{1}{2}\bar{b}$ and $C = \max(\underline{a} + 2|\underline{b}|\underline{u}^2, e^{a+|\bar{b}|\underline{u}^2})$ yields the result. $\square$

<figure><img src="image_2.png" /></figure>

## ACKNOWLEDGEMENTS

I thank Bertrand Lods with whom I had fruitful discussions and Matthieu Dolbeault who suggested to me the proof of the nice inequality (92).

I have no competing interests to declare that are relevant to the content of this article.

## REFERENCES

[1] R. Alonso, V. Bagland, L. Desvillettes, and B. Lods. About the use of entropy production for the Landau-Fermi-Dirac equation. *J. Stat. Phys.*, 183(1):Paper No. 10, 27, 2021.

[2] R. Alonso, V. Bagland, L. Desvillettes, and B. Lods. About the Landau-Fermi-Dirac equation with moderately soft potentials. *Arch. Ration. Mech. Anal.*, 244(3):779-875, 2022.

[3] R. Alonso, V. Bagland, and B. Lods. Long time dynamics for the Landau-Fermi-Dirac equation with hard potentials. *J. Differential Equations*, 270:596-663, 2021.

[4] R. Alonso, V. Bagland, and B. Lods. Long time dynamics for the landau-fermi-dirac equation with hard potentials. *Journal of Differential Equations*, 270:596-663, 2021.

[5] A. V. Bobylev and C. Cercignani. On the rate of entropy production for the Boltzmann equation. *J. Statist. Phys.*, 94(3-4):603-618, 1999.

[6] F. Bolley and C. Villani. Weighted Csiszár-Kullback-Pinsker inequalities and applications to transportation inequalities. *Ann. Fac. Sci. Toulouse Math. (6)*, 14(3):331-352, 2005.

[7] M. Breden and L. Desvillettes. Rigorous study of the equilibria of collision kernels appearing in the theory of weak turbulence. *Arch. Ration. Mech. Anal.*, 235(2):1151-1176, 2020.

[8] M. J. Cáceres, J. A. Carrillo, and J. Dolbeault. Nonlinear stability in $L^p$ for a confined system of charged particles. *SIAM J. Math. Anal.*, 34(2):478-494, 2002.

[9] María J. Cáceres, José A. Carrillo, and Jean Dolbeault. A generalization of the Csiszár-Kullback inequality. https://www.ceremade.dauphine.fr/~dolbeaul/Teaching/files/SNS-2012/Csiszar-Kullback.pdf.

[10] K. Carrapatoso, L. Desvillettes, and L. He. Estimates for the large time behavior of the Landau equation in the Coulomb case. *Arch. Ration. Mech. Anal.*, 224(2):381-420, 2017.

[11] C. Cercignani. H-theorem and trend to equilibrium in the kinetic theory of gases. *Arch. Mech. (Arch. Mech. Stos.)*, 34(3):231-241 (1983), 1982.

[12] S. Chapman and T. G. Cowling. *The Mathematical Theory of Non-uniform Gases*. Cambridge University Press, Cambridge, 1939.

[13] L. Desvillettes. Entropy dissipation estimates for the Landau equation in the Coulomb case and applications. *J. Funct. Anal.*, 269(5):1359-1403, 2015.

[14] L. Desvillettes. Structure entropique du noyau de collision de Landau. In *Séminaire Laurent Schwartz-Équations aux dérivées partielles et applications. Année 2014-2015*, pages Exp. No. XIV, 9. Ed. Éc. Polytech., Palaiseau, 2016.
[15] L. Desvillettes. Some remarks about the link between the Fisher information and Landau or Landau-Fermi-Dirac entropy dissipation. preprint. ArXiv: 2401.02877, 2024.

[16] L. Desvillettes, C. Mouhot, and C. Villani. Celebrating Cercignani's conjecture for the Boltzmann equation. *Kinet. Relat. Models*, 4(1):277-294, 2011.

[17] L. Desvillettes and C. Villani. On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. *Comm. Partial Differential Equations*, 25(1-2):261-298, 2000.

[18] J. Dolbeault. Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles. *Arch. Rational Mech. Anal.*, 127(2):101-131, 1994.

[19] G. L. Gilardoni. On Pinsker's and Vajda's type inequalities for Csiszár's *f*-divergences. *IEEE Trans. Inform. Theory*, 56(11):5377-5386, 2010.

[20] A. Jüngel. *Entropy methods for diffusive partial differential equations*, volume 804. Springer, 2016.

[21] X. Lu. A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long-time behavior. *J. Statist. Phys.*, 98(5-6):1335-1394, 2000.

[22] X. Lu. On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles. *J. Statist. Phys.*, 105(1-2):353-388, 2001.

[23] X. Lu. On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles. *J. Statist. Phys.*, 116(5):1597-1649, 2004.

[24] X. Lu and B. Wennberg. On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. *Arch. Ration. Mech. Anal.*, 168(1):1-34, 2003.

[25] L. W. Nordhiem. On the kinetic method in the new statistics and application in the electron theory of conductivity. *Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character*, 119(783):689-698, 1928.

[26] G. Toscani and C. Villani. Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. *Comm. Math. Phys.*, 203(3):667-706, 1999.

[27] G. Toscani and C. Villani. On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. *J. Statist. Phys.*, 98(5-6):1279-1309, 2000.

[28] E. A. Uehling and G. E. Uhlenbeck. Transport phenomena in Einstein-Bose and Fermi-Dirac gases. i. *Physical Review*, 43(7):552, 1933.

[29] C. Villani. A review of mathematical topics in collisional kinetic theory. In *Handbook of mathematical fluid dynamics, Vol. I*, pages 71-305. North-Holland, Amsterdam, 2002.

[30] C. Villani. Cercignani's conjecture is sometimes true and always almost true. *Comm. Math. Phys.*, 234(3):455-490, 2003.
[15] L. Desvillettes. Some remarks about the link between the Fisher information and Landau or Landau-Fermi-Dirac entropy dissipation. preprint. ArXiv: 2401.02877, 2024.

[16] L. Desvillettes, C. Mouhot, and C. Villani. Celebrating Cercignani's conjecture for the Boltzmann equation. *Kinet. Relat. Models*, 4(1):277-294, 2011.

[17] L. Desvillettes and C. Villani. On the spatially homogeneous Landau equation for hard potentials. II. H-theorem and applications. *Comm. Partial Differential Equations*, 25(1-2):261-298, 2000.

[18] J. Dolbeault. Kinetic models and quantum effects: a modified Boltzmann equation for Fermi-Dirac particles. *Arch. Rational Mech. Anal.*, 127(2):101-131, 1994.

[19] G. L. Gilardoni. On Pinsker's and Vajda's type inequalities for Csiszár's *f*-divergences. *IEEE Trans. Inform. Theory*, 56(11):5377-5386, 2010.

[20] A. Jüngel. *Entropy methods for diffusive partial differential equations*, volume 804. Springer, 2016.

[21] X. Lu. A modified Boltzmann equation for Bose-Einstein particles: isotropic solutions and long-time behavior. *J. Statist. Phys.*, 98(5-6):1335-1394, 2000.

[22] X. Lu. On spatially homogeneous solutions of a modified Boltzmann equation for Fermi-Dirac particles. *J. Statist. Phys.*, 105(1-2):353-388, 2001.

[23] X. Lu. On isotropic distributional solutions to the Boltzmann equation for Bose-Einstein particles. *J. Statist. Phys.*, 116(5):1597-1649, 2004.

[24] X. Lu and B. Wennberg. On stability and strong convergence for the spatially homogeneous Boltzmann equation for Fermi-Dirac particles. *Arch. Ration. Mech. Anal.*, 168(1):1-34, 2003.

[25] L. W. Nordhiem. On the kinetic method in the new statistics and application in the electron theory of conductivity. *Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character*, 119(783):689-698, 1928.

[26] G. Toscani and C. Villani. Sharp entropy dissipation bounds and explicit rate of trend to equilibrium for the spatially homogeneous Boltzmann equation. *Comm. Math. Phys.*, 203(3):667-706, 1999.

[27] G. Toscani and C. Villani. On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds. *J. Statist. Phys.*, 98(5-6):1279-1309, 2000.

[28] E. A. Uehling and G. E. Uhlenbeck. Transport phenomena in Einstein-Bose and Fermi-Dirac gases. i. *Physical Review*, 43(7):552, 1933.

[29] C. Villani. A review of mathematical topics in collisional kinetic theory. In *Handbook of mathematical fluid dynamics, Vol. I*, pages 71-305. North-Holland, Amsterdam, 2002.

[30] C. Villani. Cercignani's conjecture is sometimes true and always almost true. *Comm. Math. Phys.*, 234(3):455-490, 2003.