Transferring Cercignani’s conjecture-type inequalities from the classical to the fermionic Boltzmann equation

IMB seminar

January 23, 2024

Thomas Borsoni

Laboratoire Jacques-Louis Lions, Sorbonne Université

Outline

1. Quantum Boltzmann for fermions

3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

2. Relaxation to equilibrium with entropy methods

Classical Boltzmann equation

- Distribution of molecules in a rarefied gas

- Distributions of charged particles in plasmas (Landau equation)

Quantum Boltzmann for fermions in physics

- Distribution of electrons in semi-conductors

- High-energy nuclear physics

The kinetic mesoscopic approach

(homogeneous)

microscopic states

states density

\partial_t f_t = Q(f_t)

Q(f_t)(v)

f_t(v) \geq 0

interactions

interaction operator

conserved quantities

mass

momentum

energy...

\psi_0(v), \psi_1(v), ...

\displaystyle \int Q(f) \psi_i = 0

1. Quantum Boltzmann for fermions

the classical boltzmann equation

1. Quantum Boltzmann for fermions

\partial_t f_t(v) = Q_0(f_t),

(homogeneous)

\displaystyle v' = \frac{v+v_*}{2} + \frac{|v-v_*|}{2} \sigma, \qquad v'_* = \frac{v+v_*}{2} - \frac{|v-v_*|}{2} \sigma.

(B)

where

with

Q_{0}(f)(v) = \iint_{\mathbb{R}^3 \times \mathbb{S}^2} \left[f' f'_* - f f_* \right] B(v-v_*, \sigma) \, \mathrm{d} \sigma \, \mathrm{d} v_*,

Features:

assumes chaos, symmetry, reversibility
conserves of mass, momentum, energy
decrease of entropy
equilibria: Maxwellians (Gibbs)

(+ diracs)

the fermionic boltzmann equation

1. Quantum Boltzmann for fermions

Pauli's exclusion principle: occupied states are not accessible
quantum parameter \(\varepsilon \propto \frac{\hbar^3}{m^3 \beta}\), for electrons: \( \varepsilon \sim 10^{-10} \ll 1\)
equilibria = Fermi-Dirac distributions

\partial_t f_t(v) = Q_{\color{purple}\varepsilon}(f_t),

(homogeneous)

\displaystyle v' = \frac{v+v_*}{2} + \frac{|v-v_*|}{2} \sigma, \qquad v'_* = \frac{v+v_*}{2} - \frac{|v-v_*|}{2} \sigma.

(BFD)

where

with

Q_{\color{purple} \varepsilon}(f)(v) = \iint_{\mathbb{R}^3 \times \mathbb{S}^2} \left[f' f'_* \textcolor{purple}{(1 - \varepsilon f) (1-\varepsilon f_*)} - f f_* \textcolor{purple}{(1 - \varepsilon f')(1-\varepsilon f'_*)} \right] B(v-v_*, \sigma) \, \mathrm{d} \sigma \, \mathrm{d} v_*,

(+ saturated state)

Features:

assumes chaos, symmetry, reversibility
conserves of mass, momentum, energy
decrease of entropy

Properties of BFD

1. Quantum Boltzmann for fermions

\partial_t f_t(v) = Q_{\varepsilon}(f_t)

(BFD)

A priori properties of \(f_t\):

\(\displaystyle 0 \leq f_t \leq \frac{1}{\varepsilon} \),
\(f_t \in L^1_2(\mathbb{R}^3) \)

Normalisation of \(f_t\):

\int_{\mathbb{R}^3} f_t(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \mathrm{d} v = \begin{pmatrix} \rho \\ \rho u \\ 3 \rho T + \rho|u|^2 \end{pmatrix}.

\(a_{\varepsilon}\) and \(b_{\varepsilon}\) such that

\int_{\mathbb{R}^3} M_{\varepsilon}(v) \begin{pmatrix} 1 \\ v \\ |v|^2 \end{pmatrix} \mathrm{d} v = \begin{pmatrix} \rho \\ \rho u \\ 3 \rho T + \rho|u|^2 \end{pmatrix}.

"coldest" distribution at fixed \(\rho\) and \(\varepsilon\)

Equilibria:

Fermi distributions (attractor) \[M_{\varepsilon}(v) = \frac{e^{a_{\varepsilon} + b_{\varepsilon}|v-u_{\varepsilon}|^2}}{1 + \varepsilon e^{a_{\varepsilon} + b_{\varepsilon}|v-u_{\varepsilon}|^2}}\]
Saturated distribution \[F_{\varepsilon} = \frac{1}{\varepsilon} \, \mathbf{1}_{B(u,R)} \]

non-extensive overview of mathematical literature on BFD

1. Quantum Boltzmann for fermions

- Existence of solutions to homogeneous BFD for cutoff hard potentials

[Lu 2001]

Existence and uniqueness of solutions to inhomogeneous BFD for cutoff kernels

[Dolbeault 1994]

- Relaxation to equilibrium of such solutions:

either \(f_0 = F_{\varepsilon}\) or \(f_t \; \underset{t \to \infty}{\overset{L^1}{\rightharpoonup}} \; M^{f_0}_{\varepsilon}\)

Derivation of the equation from particles system (partially formal)

[Benedetto, Castella, Esposito, Pulvirenti 2007]

(review)

at which rate?

relaxation to equilibrium, entropy methods

- Relative entropy to equilibrium, general setting

- Cercignani's conjecture-type inequalities

Equilibrium and entropy

\partial_t f_t = Q(f_t)

(generically)

relative entropy to equilibrium

H(g|M^g) \geq 0

\(M^g\) depends only on conserved quantities related to \(g\)

H(g) - H(M^g) =:

Generically, \(H\) is an entropy and \(M\) is a thermodynamical equilibrium when:

\(t \mapsto H(f_t)\) is nonincreasing
\(M\) minimizes \(H\) under the conserved quantities constraints

used to quantify relaxation to equilibrium

not necessarily the only equilibria of the dynamic

2. Relaxation to equilibrium, entropy methods

Entropy dissipation and entropy methods

\partial_t f_t = Q(f_t)

Entropy dissipation \(D\)

\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} H(f_t) = - D(f_t)

\(D \) non-negative operator

Try to prove \(D(f) \gtrsim H(f|M^{f})^{1+\delta}\)

(functional inequality)

Entropy method

To obtain \(H(f_t|M^{f_0}) \lesssim t^{-1/\delta}\)

\displaystyle \frac{\mathrm{d}}{\mathrm{d} t} H(f_t|M^{f_0}) = - D(f_t)

Try to prove \(D(f) \gtrsim H(f|M^{f})\)

To obtain \(H(f_t|M^{f_0}) \lesssim e^{-Ct} \)

(Grönwall)

2. Relaxation to equilibrium, entropy methods

Various entropies and equilibria

classical

Fermionic

Maxwellian distribution

Fermi distribution

Fermi entropy

Boltzmann entropy

H_0(g) = \int g \log g - g

H_{\varepsilon}(f) = \int f \log f + \varepsilon^{-1}\int (1 - \varepsilon f) \log (1-\varepsilon f)

M_0(v) = \rho (2\pi T)^{-3/2} \, \exp \left(- \frac{|v-u|^2}{2T} \right)

M_{\varepsilon}(v) = \frac{e^{a_{\varepsilon} + b_{\varepsilon}|v-u|^2}}{1 + \varepsilon e^{a_{\varepsilon} + b_{\varepsilon}|v-u|^2}}

H_0(g) = \int \Phi_0(g)

H_{\varepsilon}(f) = \int \Phi_{\varepsilon}(f)

M_0(v) = (\Phi_0')^{-1} \left(\alpha + \beta \cdot v + \gamma |v|^2 \right)

M_{\varepsilon}(v) = (\Phi_{\varepsilon}')^{-1} \left(\alpha + \beta \cdot v + \gamma |v|^2 \right)

\Phi_0' = \log

\Phi_{\varepsilon}'(x) = \log \left(\frac{x}{1 - \varepsilon x} \right)

2. Relaxation to equilibrium, entropy methods

Cercignani's conjecture type results

D_0(g) \gtrsim H_0(g|M_0^g)^{1 + \delta}

classical Boltzmann

D_{\varepsilon}(f) \gtrsim H_{\varepsilon}(f|M^{f}_{\varepsilon})^{1+\delta}

Fermionic Boltzmann

Toscani, Villani

D^L_0(g) \gtrsim H_0(g|M_0^g)^{1 + \delta}

classical Landau

D^L_{\varepsilon}(f) \gtrsim H_{\varepsilon}(f|M^{f}_{\varepsilon})^{1+\delta}

Fermionic LAndau

Desvillettes, Villani

Alonso, Bagland Desvillettes, Lods

2. Relaxation to equilibrium, entropy methods

Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

- A bridge between the classical and the fermionic cases

- Cercignani's conjecture-type results for the fermonic Boltzmann equation

D^B_{\varepsilon}(f) \geq \kappa_0^4 \, D^B_{0}\left(\varphi_{\varepsilon}(f)\right).

an interesting link

and its implications on the entropy dissipation

\displaystyle \varphi_{\varepsilon} (x) = \frac{x}{1 - \varepsilon x},

Q_{\varepsilon}(f)(v) = \iint_{\mathbb{R}^3 \times \mathbb{S}^2} \left[f' f'_* (1 - \varepsilon f) (1-\varepsilon f_*) - f f_* (1 - \varepsilon f')(1-\varepsilon f'_*) \right] B(v-v_*, \sigma) \, \mathrm{d} \sigma \, \mathrm{d} v_*,

Q_{\varepsilon}(f)(v) = \iint_{\mathbb{R}^3 \times \mathbb{S}^2} \left[\frac{f'}{1-\varepsilon f'} \frac{f'_*}{1-\varepsilon f'_*} - \frac{f}{1-\varepsilon f} \frac{f_*}{1-\varepsilon f_*} \right] (1 - \varepsilon f) (1-\varepsilon f_*)(1 - \varepsilon f')(1-\varepsilon f'_*) B \, \mathrm{d} \sigma \, \mathrm{d} v_*,

Q_{\varepsilon}(f)(v) = \iint_{\mathbb{R}^3 \times \mathbb{S}^2} \left[\varphi_{\varepsilon}(f)' \varphi_{\varepsilon}(f)'_* - \varphi_{\varepsilon}(f) \varphi_{\varepsilon}(f)_* \right] B_{\varepsilon,f}(v-v_*, \sigma) \, \mathrm{d} \sigma \, \mathrm{d} v_*

= Q_0^{B_{\varepsilon,f}}(\varphi_{\varepsilon}(f))(v),

\displaystyle B_{\varepsilon,f}(v,v_*,\sigma) = (1 - \varepsilon f) (1-\varepsilon f_*)(1 - \varepsilon f')(1-\varepsilon f'_*) B(v-v_*, \sigma).

If \(1- \varepsilon f \geq \kappa_0 \), then \(B_{\varepsilon,f} \geq \kappa_0^4 B\) and

\implies D_{\varepsilon}^B(f) = D_{0}^{B_{\varepsilon,f}}\left(\varphi_{\varepsilon}(f)\right)

D_{\varepsilon}(f) \geq \kappa_0^4 \, D_{0}\left(\varphi_{\varepsilon}(f)\right).

positivity, symmetry, micro-reversibility

3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

If \(1- \varepsilon f > 0 \)

Our strategy

(transfer trick)

D_0(g) \gtrsim H_0(g|M_0^g)^{1 + \delta}

D_{\varepsilon}(f)

H_{\varepsilon}(f|M^f_{\varepsilon})^{1 + \delta}

we know:

we want to prove:

D_{\varepsilon}(f) \gtrsim D_{0}\left(\frac{f}{1 - \varepsilon f}\right)

\gtrsim \; \; D_{0}\left(\frac{f}{1 - \varepsilon f}\right)

\gtrsim

entropy inequality for classical Boltzmann

D_{\varepsilon}(f) \gtrsim H_{\varepsilon}(f|M^f_{\varepsilon})^{1 + \delta}

Fermi dissipation of \(f\)

\( \gtrsim\) classical dissipation of \( \displaystyle \frac{f}{1-\varepsilon f} \)

\gtrsim \; \; H_0 \left(\frac{f}{1- \varepsilon f} \left|M_0^{\frac{f}{1- \varepsilon f}} \right. \right)^{1 + \delta}

entropy inequality for fermionic Boltzmann

3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

Comparison of relative entropies

H_{0}\left(\left.\frac{f}{1 - \varepsilon f}\right|M^{\frac{f}{1 - \varepsilon f}}_{0}\right) \geq H_{\varepsilon}(f|M^f_{\varepsilon}).

H: whenever all terms make sense

Classical relative entropy to equilibrium of \(\displaystyle \frac{f}{1-\varepsilon f}\)

Fermi relative entropy to equilibrium of \(f\)

Theorem.

f \in L^1_2(\R^3)

\frac{f}{1 - \varepsilon f} \in L^1_2(\R^3) \cap L \log L(\R^3),

\displaystyle 0 \leq f < \frac{1}{\varepsilon}

For all

such that

and

[T.B.]

3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

Proof

main steps

R_g(\varepsilon) = H_{\varepsilon}\left( \varphi_{\varepsilon}^{-1}(g) \left|M^{\varphi_{\varepsilon}^{-1}(g)}_{\varepsilon} \right. \right).

Let, for \(\varepsilon \geq 0\) and \( g \in L^1_2(\R^3) \, \cap L \log L(\R^3)\),

Then \(R_g\) is decreasing on \(\R_+\).

Proposition.

Key elements of the proof:

Taylor representation of relative entropy (general form)
differentiation of \(R_g\) in \(\varepsilon\)
"magical" cancellations due to the general links entropy/equilibria

Other technicalities:

differentiability of \(R_g \) on \(\R_+^*\)
continuity of \(R_g \) at \(0\)

general considerations

specific use of Fermi-Dirac features

\bullet \; \varphi_{\varepsilon}(x) = \frac{x}{1 - \varepsilon x}

3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

R_g(0) = H_{0}\left(\left.\frac{f}{1 - \varepsilon f}\right|M^{\frac{f}{1 - \varepsilon f}}_{0}\right)

\bullet \; g = \varphi_{\varepsilon}(f)

R_g(\varepsilon) =H_{\varepsilon}(f|M^f_{\varepsilon})

Conclusion

D_0(g) \geq C_g \, H_0(g|M_0^g)^{1 + \delta}

D_{\varepsilon}(f) \geq \kappa_0^4 \, C_{\frac{f}{1-\varepsilon f}} H_{\varepsilon}(f|M_{\varepsilon}^f)^{1 + \delta}

entropy inequality for classical Boltzmann

entropy inequality for fermionic Boltzmann

(1 - \varepsilon f \geq \kappa_0)

counter-example for classical Boltzmann (Bobylev, Cercignani)

counter-example for fermionic Boltzmann

3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

Implications for fermionic Boltzmann and fermionic Landau

3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer

Fermionic Landau

Fermionic Boltzmann

D^L_{\varepsilon}(f) \geq C_{f} \, H_{\varepsilon}(f|M_{\varepsilon}^f)

new result

known result, simpler proof

D^L_{\varepsilon}(f) \geq C_{f,\delta} \, H_{\varepsilon}(f|M_{\varepsilon}^f)^{1 + \delta}

Over-Maxwellian and hard potentials

Soft potentials

Exponential convergence to equilibrium

polynomial convergence to equilibrium

D_{\varepsilon}(f) \geq C_{f} \, H_{\varepsilon}(f|M_{\varepsilon}^f)

Super-quadratic kernels

exponential convergence to equilibrium

D_{\varepsilon}(f) \geq C_{f,\delta} \, H_{\varepsilon}(f|M_{\varepsilon}^f)^{1 + \delta}

General kernels (with Maxwellian lower-bound assumption)

polynomial convergence to equilibrium

current work

D_{\varepsilon}(f) \geq \textcolor{blue}{C_{f,\delta}} \, H_{\varepsilon}(f|M_{\varepsilon}^f)^{1 + \delta}

Fermionic Boltzmann, hard potentials with cutoff

Proof of polynomial convergence to equilibrium

Requirements to apply \((\ast)\)

prove Maxwellian lower-bound
Control \(\textcolor{blue}{C_{f,\delta}}\): prove \(1- \varepsilon f \geq \kappa_0\) and control moments

(\ast)

In collaboration with B. Lods

Thank you for your attention!