Transfer of entropy inequalities from the classical to the fermionic Boltzmann equation
Lions Magenes days 2024
December 18, 2024
Thomas Borsoni
Laboratoire Jacques-Louis Lions
Context
Distribution of electrons in semi-conductors
Boltzmann-Fermi-Dirac equation
In physics:
Evolution equation:
\partial_t f = Q(f)
(homogeneous)
Existence of equilibrium state
(Boltzmann for fermions)
Motivation
Explicit rate of relaxation to equilibrium for sol. to Boltzmann-Fermi-Dirac eq.
Main result
entropy ineq.
classical molecules
entropy ineq.
fermions
Means
Entropy methods
-> functional inequalities
\implies
known
new!
Outline
1. Quantum Boltzmann for fermions
3. Transfer of entropy inequalities
2. Entropy methods
the classical boltzmann equation
1. Quantum Boltzmann for fermions
\partial_t f_t(v) = Q_0(f_t)(v),
(homogeneous)
(B)
where
Q_{0}(f)(v) = \int \left[f(v') f(v'_*) - f(v) f(v_*) \right]
B
Features:
- symmetry, reversibility
- conserves of mass, momentum, energy
- decrease of classical entropy
v
v_*
v'
v'_*
v
Equilibrium: Maxwellian
M_0
the boltzmann-Fermi-Dirac equation
1. Quantum Boltzmann for fermions
- quantum parameter \(\varepsilon \propto \hbar^3\)
\partial_t f_t(v) = Q_{\color{purple}\varepsilon}(f_t)(v),
(homogeneous)
(BFD)
where
Q_{\color{purple} \varepsilon}(f)(v) = \int \left[f' f'_* \textcolor{purple}{(1 - \varepsilon f) (1-\varepsilon f_*)} - f f_* \textcolor{purple}{(1 - \varepsilon f')(1-\varepsilon f'_*)} \right]
B
Features:
- symmetry, reversibility
- conserves of mass, momentum, energy
- decrease of Fermi entropy
Equilibrium: Fermi-Dirac statistics
(+ saturated state)
- \(\displaystyle 0 \leq f_t \leq \frac{1}{\varepsilon} \)
Boltzmann + Pauli's exclusion principle
M_{\varepsilon}
relaxation to equilibrium, entropy methods
- Relative entropy to equilibrium, general setting
- Entropy inequalities results
Equilibrium and entropy
\partial_t f_t = Q(f_t)
(generically)
relative entropy to equilibrium
H(g|M^g) \geq 0
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 some constraints (mass, energy...)
used to quantify distance to equilibrium
2. Relaxation to equilibrium, entropy methods
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
Entropy inequalities 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}
Boltzmann-FERMI-DIRAC
Toscani, Villani
?
2. Relaxation to equilibrium, entropy methods
\(D_0\): classical entropy dissipation
\(H_0\): classical entropy
\(D_\varepsilon\): Fermi-Dirac entropy dissipation
\(H_\varepsilon\): Fermi-Dirac entropy
Obtention of entropy inequalities for Boltzmann-Fermi-Dirac
- Transfer of ineq. from classical to quantum
- Application to the Boltzmann-Fermi-Dirac equation
a first remark
If \(1- \varepsilon f \geq \kappa_0 \), then
\displaystyle D_{\varepsilon}(f) \gtrsim D_{0}\left( \frac{f}{1 - \varepsilon f} \right).
3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer
\kappa_0 \in (0,1)
Fermi-Dirac entropy dissipation of \(f\)
Classical entropy dissipation of \(\displaystyle \frac{f}{1-\varepsilon f}\)
Transfer of inequalities
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-Dirac 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
(Toscani, Villani)
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
3. Relaxation to equilibrium for fermionic Boltzmann: a method of transfer
Proof
R_g(\varepsilon) = H_{\varepsilon}\left( \varphi_{\varepsilon}^{-1}(g) \left|M^{\varphi_{\varepsilon}^{-1}(g)}_{\varepsilon} \right. \right).
Let, for \(\varepsilon \geq 0\),
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
\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)
g = \frac{f}{1 - \varepsilon f},
R_g(\varepsilon) =H_{\varepsilon}(f|M^f_{\varepsilon})
take
\geq
then
\&
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
Convergence to equilibrium
Proof of algebraic convergence to equilibrium
with B. Lods
\|f_t-M_{\varepsilon}\|_{L^p_k} \leq C \, (1+t)^{- \eta/p}
\(p \in [1,\infty), \, k \geq 0\), and with \(C,\eta\) explicit.
(Boltzmann-Fermi-Dirac homogeneous cut-off hard potentials)
Thank you for your attention!
Lions Magenes
By Thomas Borsoni
