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

v
\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

Taylor representation  of relative entropy to equilibrium

(entropy)

(conserved quantities)

\displaystyle \int M^f \, \psi_i \, \mathrm{d} v = \int f \, \psi_i \, \mathrm{d} v.

(equilibrium)

\displaystyle H(f|M^f) = \int_0^1 (1-\tau) \int(f - M^f)^2 \, \Phi''(M^f + \tau (f-M^f)) \, \mathrm{d} v \, \mathrm{d} \tau

Proposition.

  • consider \(\psi_0, \psi_1, \dots, \psi_n\) conserved quantities
  • consider \(\Phi \in \mathcal{C}^2(\mathbb{R})\)  strictly convex, and \( \displaystyle H(f) = \int \Phi(f) \, \mathrm{d} v\)
  • assume \(\exists \, M^f = (\Phi')^{-1}(\alpha_0 \psi_0 + \dots + \alpha_n \psi_n) \)  such that  

Then

Consider a distribution \( f\).

\displaystyle H(f|M^f) = \int \, \int_{M^f(v)}^{f(v)} \, (f(v) - x) \, \Phi''(x) \, \mathrm{d} x \, \mathrm{d} v

Remark: suited to obtain general Cszisar-Kullback inequalities

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 \((\ast)\)

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

In collaboration with B. Lods

Perspectives

general results on entropies

Conserved quantities \(\displaystyle \psi_1, \psi_2, \dots\)

Equilbrium \(\displaystyle M = (\Phi')^{-1}(\alpha_1 \psi_1 + \alpha_2 \psi_2 + \dots)\)

Entropy \(\displaystyle H(f) = \int \Phi(f)\)

  • Taylor representation of relative entropy to equilibrium
\displaystyle H(f|M^f) = \int_0^1 (1-\tau) \int(f - M^f)^2 \, \Phi''(M^f + \tau (f-M^f)) \, \mathrm{d} v \, \mathrm{d} \tau
  • General Cszisar-Kullback inequalities (\(1 \leq p \leq 2\))
\displaystyle \left\| f - M^f \right\|_{L^p_{\varpi}}^2 \leq C_{f,\Phi,p, \varpi} \, H(f|M^f)
  • Comparison of relative entropies to equilibrium \((\Phi_{\lambda})_{\lambda}\), study of \(R_f(\lambda) \)

Analogous results (less useful) for bosonic Boltzmann

Thank you for your attention!