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 \((\ast)\)
- 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
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!
Bordeaux presentation
By Thomas Borsoni
