gaz et plasmas ​🌬️​

chaos et fourmis 🐜​​

Cermics

Oct 14, 2024

Thomas Borsoni

postdoc \(1^{ere}\) année

ma thèse

mon postdoc

Contributions around the Boltzmann equation and some of its variants

supervisé par

Laurent Boudin & Laurent Desvillettes

au Laboratoire Jacques-Louis Lions

Sorbonne Université

ma thèse :

Mesoscopic

Microscopic

Macroscopic

(statistical)

description of gases

Density of molecules:   \(f \equiv f_{t,x}(v)\)

The original Boltzmann equation

Statistical description of a monoatomic gas

[A. Greg: Kinetic theory of gases, wikipedia.]

\partial_t f_{t,x}(v) + v \cdot \nabla_x f_{t,x}(v) = Q(f_{t,x})(v)
v
x
+

advection

collisions

(e.g. \(\mathrm{Ar} \) )

[A. Greg: Kinetic theory of gases, wikipedia.]

Focus on collisions

the Homogeneous Boltzmann equation

Density of molecules:   \(f_t(v)\)

\partial_t f_t(v) = Q(f_t)(v)

\(x \)

advection

\(+ \, v \cdot \nabla_x f\)

Entropy and equilibrium

v
f^{\rm in}
f_t \underset{t \to \infty}{\longrightarrow} c \, e^{-\frac{|v-\textcolor{blue}{u}|^2}{2 \textcolor{blue}{T}} }
v
\textcolor{blue}{u}
\textcolor{blue}{T}

The Boltzmann entropy:

\newcommand{\dd}{\mathrm{d}} H (f) := \int_{\R^3} (f \log f - f)(v) \, \dd v

2.  \(D(g) = 0 \iff g =M \) a Maxwellian:

characterization of equilibria

M(v) \propto \exp\left(-\frac{|v-\textcolor{blue}{u}|^2}{2 \textcolor{blue}{T}} \right)
\partial_t f_t(v) = Q(f_t)(v), \\ f_0 = f^{\rm in},

Boltzmann's H Theorem

1. If \(f \equiv f_t(v)\) solves

\(2^{nd}\) principle of thermodynamics

\((HB)\)

then

\newcommand{\dd}{\mathrm{d}} \frac{\dd}{\dd t}H (f_t) =: - D(f_t) \leq 0.

original

monoatomic molecules

 

          polyatomic molecules

 

Part \(\mathrm{I}.1\)

Part \(\mathrm{ I}.2\)

polyatomic with

resonant collisions

fermions

Part \(\mathrm{II}\)

e.g. \(\mathrm{Ar}\)

e.g. \(\mathrm{N_2}\), \(\mathrm{H_2O}\)

e.g. \(\mathrm{CO_2}\)

e.g. \(\mathrm{e^-}\)

My work

- Modelling

- Analysis

- Simulations

- Functional analysis

- PDEs

Transfer of entropy inequalities 

from Boltzmann to BOLTZMANN-Fermi-Dirac

(Partie \(\mathrm{II}\) de ma thèse)

Distribution of electrons in semi-conductors

The Boltzmann-Fermi-Dirac equation

Boltzmann for fermions

the Boltzmann equation

\partial_t f_t(v) = Q_{\textcolor{green}{0}}(f_t)(v),

(homogeneous)

conservation of mass, momentum, energy

v
v_*
v'
v'_*

\(f \equiv f_t(v)\) density of molecules

v
\newcommand{\dd}{\mathrm{d}} \textcolor{black}{Q_{\textcolor{green}{0}}(f)(v) = }\iint_{\R^3 \times \mathbb{S}^2} \textcolor{black}{\big( f(v') f(v'_*) - f(v) f(v_*) \big)} \, B(v,v_*,\sigma) \, \dd \sigma \, \dd v_*

1. Dissipation of the Boltzmann entropy

2. Equilibria: Maxwellian (Gibbs) distribution

H Theorem:

v
\begin{align*} v' = \frac{v+v_*}{2} + \frac{|v-v_*|}{2} \sigma \\ v'_* = \frac{v+v_*}{2} - \frac{|v-v_*|}{2} \sigma\\ \end{align*}
\partial_t f_t(v) = Q_{\textcolor{purple}{\delta}}(f_t)(v)
v

the Boltzmann-Fermi-Dirac equation

Q_{\color{purple} \delta}(f)(v) = \iint_{\R^3 \times \mathbb{S}^2} \left[f' f'_* \textcolor{purple}{(1 - \delta f) (1-\delta f_*)} - f f_* \textcolor{purple}{(1 - \delta f')(1-\delta f'_*)} \right] B \; \mathrm{d} v_* \, \mathrm{d} \sigma
0 \leq f_{t} \leq \frac{1}{\textcolor{purple}{\delta}}
v
v_*
v'
v'_*

Fermions -> Pauli exclusion principle -> quantum parameter \(\delta>0\) :

1. Dissipation of the Fermi-Dirac entropy

2. Equilibria: Fermi-Dirac statistics

H Theorem:

(+ saturated state:                )

\frac{1}{\textcolor{purple}{\delta}}
v

(homogeneous ​)

conservation of mass, momentum, energy

\begin{align*} v' = \frac{v+v_*}{2} + \frac{|v-v_*|}{2} \sigma \\ v'_* = \frac{v+v_*}{2} - \frac{|v-v_*|}{2} \sigma\\ \end{align*}
\newcommand{\Sb}{\mathbb{S}} \sigma \in \Sb^2

additional feature

\(f \equiv f_t(v)\) density of fermions

Fermi-Dirac entropy

Boltzmann entropy

H_{\textcolor{green}{0}}(g) = \int g \log g - g
H_{\textcolor{purple}{\delta}}(f) = \int f \log f + {\textcolor{purple}{\delta}}^{-1} (1 - {\textcolor{purple}{\delta}} f) \log (1-{\textcolor{purple}{\delta}} f)
M_{\textcolor{green}{0}}(v) = e^{a - b|v-u|^2}
\displaystyle M_{\textcolor{purple}{\delta}}(v) = \frac{e^{a - b|v-u|^2}}{1 + \textcolor{purple}{\delta} e^{a - b|v-u|^2}}

Equilibrium: Fermi-Dirac statistics

Equilibrium: Maxwellian

entropies and equilibria

Boltzmann

Boltzmann-Fermi-Dirac

Fermi-Dirac entropy

H_{\textcolor{green}{0}}(g) = \int \Phi_{\textcolor{green}{0}}(g)
H_{\textcolor{purple}{\delta}}(f) = \int \Phi_{\textcolor{purple}{\delta}}(f)
M_{\textcolor{green}{0}}(v) = (\Phi_{\textcolor{green}{0}}')^{-1} \left(\alpha + \beta \cdot v + \gamma |v|^2 \right)
\displaystyle M_{\textcolor{purple}{\delta}}(v) = (\Phi_{\textcolor{purple}{\delta}}')^{-1} \left(\alpha + \beta \cdot v + \gamma |v|^2 \right)

Equilibrium: Fermi-Dirac statistics

Equilibrium: Maxwellian

entropies and equilibria

Boltzmann

Boltzmann-Fermi-Dirac

\Phi_{\textcolor{purple}{\delta}}'(x) = \log \left(\frac{x}{1 - \textcolor{purple}{\delta} x} \right)
\Phi_{\textcolor{green}{0}}' = \log

entropy    \(\displaystyle H : h \mapsto \int \Phi(h)\)          \(\Phi\)    \(\mathcal{C}^2\) st. convex

equilibrium 

M = (\Phi')^{-1} (\alpha \, \textcolor{blue}{\text{mass}} + \beta \cdot \textcolor{blue}{\text{momentum}} + \gamma \, \textcolor{blue}{\text{energy}})

Boltzmann entropy

\partial_t f_t = Q(f_t)

\(D(g) \gtrsim {H(g|M^g)}^{1+\alpha}\)

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

\(D(g) \geqslant C H(g|M^g)\)

the entropy method

Relative entropy to equilibrium:

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

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

Csiszár-Kullback-Pinsker

\|f_t - M^{f_0}\|_{L^1}^2 \lesssim H(f_t|M^{f_0})
H(g|M^g) := H(g) - H(M^g) \geq 0
D_{\textcolor{green}{0}}(g) \gtrsim H_{\textcolor{green}{0}}(g|M_{\textcolor{green}{0}}^g)^{1 \textcolor{grey}{+ \alpha}}

Boltzmann

Toscani, Villani

Landau

D_{\textcolor{purple}{\delta}, land}(f) \gtrsim H_{\textcolor{purple}{\delta}}(f |M_{\textcolor{purple}{\delta}}^f)^{1 \textcolor{grey}{+ \alpha}}

LAndau-Fermi-Dirac

Desvillettes, Villani

Alonso, Bagland Desvillettes, Lods

D_{\textcolor{purple}{\delta}}(f) \gtrsim H_{\textcolor{purple}{\delta}}(f|M_{\textcolor{purple}{\delta}}^f)^{1 \textcolor{grey}{+ \alpha}}

Boltzmann-FERMI-DIRAC

known entropy inequalities

\(D_{\textcolor{green}{0}, land}(g) \gtrsim H_{\textcolor{green}{0}}(g|M_{\textcolor{green}{0}}^g)^{1 + \textcolor{grey}{\alpha}}\)

c'est moi

Main IDea

\forall \, g \in \mathcal{G}, \qquad D_{\textcolor{green}{0}}(g) \gtrsim H_{\textcolor{green}{0}}(g|M_{\textcolor{green}{0}}^g)^{1 + \alpha}

entropy inequality for Boltzmann

\forall \, f \in \mathcal{F}, \qquad D_{\textcolor{purple}{\delta}}(f) \gtrsim H_{\textcolor{purple}{\delta}}(f|M_{\textcolor{purple}{\delta}}^f)^{1 + \alpha}

entropy inequality for Boltzmann-Fermi-Dirac

\implies

known

new!

(Toscani, Villani)

transfer of inequalities

D_{\textcolor{green}{0}}(g) \gtrsim H_{\textcolor{green}{0}}(g|M_{\textcolor{green}{0}}^g)^{1 + \alpha}

We know:

\gtrsim \; \; D_{\textcolor{green}{0}}\left(\frac{f}{1 - \textcolor{purple}{\delta} f}\right)

entropy inequality for Boltzmann

\gtrsim \; \; H_{\textcolor{green}{0}} \left(\frac{f}{1- \textcolor{purple}{\delta} f} \left| M_{\textcolor{green}{0}}^{\frac{f}{1- \textcolor{purple}{\delta} f}} \right. \right)^{1 + \alpha}

entropy inequality for Boltzmann-Fermi-Dirac

We want:

D_{\textcolor{purple}{\delta}}(f) \gtrsim D_{\textcolor{green}{0}}\left(\frac{f}{1 - \textcolor{purple}{\delta} f}\right)

Fermi-Dirac dissipation of \(f\)

Boltzmann dissipation of \( \displaystyle \frac{f}{1-\textcolor{purple}{\delta} f} \)

\( \gtrsim\)

D_{\textcolor{purple}{\delta}}(f) \textcolor{red} \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \textcolor{red}{\gtrsim} \;H_{\textcolor{purple}{\delta}}(f|M_{\textcolor{purple}{\delta}}^f)^{1 + \alpha}

?

H_{\textcolor{green}{0}}\left(\left.\frac{f}{1 - \textcolor{purple}{\delta} f}\right|M^{\frac{f}{1 - \textcolor{purple}{\delta} f}}_{\textcolor{green}{0}}\right) \geqslant H_{\textcolor{purple}{\delta}}(f|M^f_{\textcolor{purple}{\delta}}).

as soon as all terms make sense

Boltzmann relative entropy to equilibrium of  \(\displaystyle \frac{f}{1-\textcolor{purple}{\delta} f}\)

Fermi-Dirac relative entropy to equilibrium of \(f\)

Theorem 1.

f \in L^1_2(\R^3)
\frac{f}{1 - \delta f} \in L^1_2(\R^3) \cap L \log L(\R^3),
\displaystyle 0 \leqslant f < \frac{1}{\delta}

For all

such that

and

\delta > 0

and

comparison of relative entropies

[B.]

R_g : \delta \in \R_+ \mapsto H_{\delta}\left( \frac{g}{1 + \delta g} \left|M^{\frac{g}{1 + \delta g}}_{\delta} \right. \right).

Let

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

Proposition.

0 \leq g \in L^1_2(\R^3) \cap L \log L(\R^3)

and

Proof of the theorem

proof of the proposition

Key elements:

  • Taylor representation of the relative entropy to eq.
  • general link between entropy and equilibrium
  • fact that \(x \mapsto -\left(\frac{x}{1+\delta x} \right)^2\) is decreasing

Other technicalities:

  • differentiability on \(\R_+^*\)
  • continuity at \(\delta = 0\)

general considerations

specific use of Fermi-Dirac features

Show \(R_g' \leq 0\) on \(\R_+^*\) and \(R_g\)   \(\mathcal{C}^0\) on \(\R_+ \)

Relaxation to equilibrium for Boltzmann-fermi-dirac

\|f^{\delta}_t-M_{\delta}\|_{L^p_k} \leq C \, (1+t)^{- \eta/p},

\(p \in [1,\infty), \, k \geq 0\), and with \(C,\eta\)  explicit and uniform in \(\delta \leqslant \delta^{\rm in}\).

Theorem 2.

with Lods

Let \(0\leqslant f^{\rm in} \in L^1_3(\R^3)\). Then  \(\exists \delta^{\rm in} > 0\) such that  \(\forall \delta \in (0,\delta^{\rm in})\), if \(f^{\delta} \) sol. to Boltzmann-Fermi-Dirac w/ cut-off hard potentials,

(\(\delta\) is the quantum parameter)

then

\|f^{\delta}_t-M_{\delta}\|_{L^p_k} \leq C \, (1+t)^{- \eta/p},

\(p \in [1,\infty), \, k \geq 0\), and with \(C,\eta\)  explicit and uniform in \(\delta \leqslant \delta^{\rm in}\).

Relaxation to equilibrium for Boltzmann-fermi-dirac

Theorem 2.

[B., Lods]

Let \(0\leqslant f^{\rm in} \in L^1_3(\R^3)\). Then  \(\exists \delta^{\rm in} > 0\) such that  \(\forall \delta \in (0,\delta^{\rm in})\), if \(f^{\delta} \) sol. to Boltzmann-Fermi-Dirac w/ cut-off hard potentials,

(\(\delta\) is the quantum parameter)

Proof's core ingredients:

  1. \(L^{\infty}\)-bound on \(f^{\delta}\) independent of \(\delta\)
  2. Entropy/entropy production inequality
  3. Control of moments
  4. Maxwellian lower-bound
  5. Csiszar-Kullback-Pinsker inequality
1 - \delta f^{\delta} \geq \kappa
D_{\delta}(f_t^{\delta}) \geqslant C_t \; H_{\delta}(f_t^{\delta}|M_{\delta})^{1 + \alpha}
C_t \geqslant \widetilde{C}
\|f^{\delta}_t-M_{\delta}\|^2_{L^1_k} \leqslant C'\, H_{\delta}(f_t^{\delta}|M_{\delta})

then

\|f-M^f\|^2_{L^p_{\varpi}} \leqslant {\small C_{\Phi, \varpi, p, f, M^f}} \; H(f|M^f),

Proposition.

(general entropy)

\(\displaystyle H(f) = \int\Phi(f)\),   \(\Phi \; \; \mathcal{C}^2\) st. convex,   \(M^f\) equilibrium, and

C_{\Phi, \varpi, p,f,M^f} = \left(\int_0^1 (1-\tau) \left\| \Phi''((1-\tau)M^f + \tau f)^{-1} \right\|_{L^{\frac{p}{2-p}}_{\varpi^2}}^{-1} \, \mathrm{d} \tau \right)^{-1}

For any \(f\), any \(p \in [1,2]\) and \(\varpi\) weight,

\|f-M_0^f\|^2_{L^p_{\varpi}} \leqslant 2 \max \left(\|f\|_{L^{\frac{p}{2-p}}_{\varpi^2}}, \|M_0^f\|_{L^{\frac{p}{2-p}}_{\varpi^2}} \right)\, H_{0}(f|M_0^f),

Corollary.

For any \(0\leq f \in L^1_2\cap L \log L(\R^3)\), \(p \in [1,2]\) and \(\varpi : \R^3 \to \R_+\),

(Boltzmann entropy)

\(\displaystyle H_0\) Boltzmann entropy and \(M_0^f\) Maxwellian.

[simplified]

Bonus: extended Csiszàr-Kullback-Pinsker inequalities

chaos et fourmis 🐜​​

mon postdoc :

Cermics

Part \(\mathrm{I}\):

Part \(\mathrm{II}\):

Propagation of chaos, mean-field limits for SPDEs

- Proba

- Analysis

- SPDEs

Traffic behavior in Argentine ants:

Recovering microscopic model from experiments

- Modelling (micro / meso)

- Parameter estimation

- Simulations

Merci pour votre attention !

🐜​🐜​🐜​🐜​🐜​🐜​🐜​🐜​🐜​🐜​🐜​

Proposition.

0\leqslant f \in L^1_2(\R^3)
\displaystyle \textcolor{black}{ 1 - \textcolor{purple}{\delta} f \geqslant} \kappa,

For all

such that

\textcolor{purple}{\delta} \textcolor{black}{> 0}, \;\; \kappa \textcolor{black}{\in (0,1)}

and

Classical / Fermi-Dirac equivalence

\textcolor{black}{H_{\textcolor{purple}{\delta}}(f|M^f_{\textcolor{purple}{\delta}}) \leqslant H_{\textcolor{green}{0}}\left(\left.\frac{f}{1 - \textcolor{purple}{\delta} f}\right|M^{\frac{f}{1 - \textcolor{purple}{\delta} f}}_{\textcolor{green}{0}}\right) \leqslant} \textcolor{black}{e}^{ \textcolor{black}{16} \, { \textcolor{black}{(}\kappa^{\textcolor{black}{-1}} \textcolor{black}{- 1)} } } \textcolor{black}{\cdot H_{\textcolor{purple}{\delta}}(f|M^f_{\textcolor{purple}{\delta}})} \phantom{\frac{1-\kappa}{\kappa}}
\textcolor{black}{D_{\textcolor{purple}{\delta}}(f) \leqslant D_{\textcolor{green}{0}}\left(\frac{f}{1 - \textcolor{purple}{\delta} f}\right) \leqslant} \kappa^{\textcolor{black}{-4}} \textcolor{black}{ \cdot D_{\textcolor{purple}{\delta}}(f)}

For Boltzmann/BFD (& Landau/LFD) dissipations:

entropy inequality for Boltzmann

entropy inequality for Boltzmann-Fermi-Dirac

1 - \delta f \geq \kappa
\iff
1
\textcolor{purple}{\delta} f
\kappa

[TB]

\implies

\(\R^3 \times \textcolor{orange}{\R^3 \times \N \times \N \times \N}\)

\(\mathrm{I}.1\). internal structure of polyatomic molecules: rotation, vibration,...

v
\omega
n_1
n_2
n_3
(v,\textcolor{orange}{\omega, n_1, n_2, n_3})
\in

state of the molecule

space of states

energy of the molecule

\frac12 |v|^2 + \textcolor{orange}{\frac12 |\omega|^2 + n_1 E_1 + n_2 E_2 + n_3 E_3 + E_{pot}}

An example:

\(\R^3 \times \textcolor{orange}{\mathcal{E}}\)

\(\mathrm{I}.1\). internal structure of polyatomic molecules: general setting

v
(v,\textcolor{orange}{\zeta})
\in

state of the molecule

space of states

energy of the molecule

\frac12 |v|^2 + \textcolor{orange}{\varepsilon(\zeta)}
\newcommand{\E}{\mathcal{E}}
\zeta

\((\mathcal{E}, \mu)\)

\(\varepsilon : \mathcal{E} \to \R\)

2. Internal energy function:

existence of fundamental energy level

finiteness of the partition function

\(\bar{\varepsilon} := \varepsilon - \inf_{\mu}  {\varepsilon}\)             (\( : \mathcal{E} \to \R_+\))

1. Space of internal states:

3 points of view:  parallel with probability theory

polyatomic internal structure

probability theory setting

\( (\Omega, \; \mathbb{P}) \) space of events

\( X : \Omega \to \R \)    real random var.

\( (\mathcal{E}, \; \mu) \) space of internal states

\( \bar{\varepsilon} : \mathcal{E} \to \R_+ \)    energy function

\( (\R, \, \mathbb{P}_X)\) space of outcomes

\( \mathbb{P}_X\) on \(\R\)    law of \(X\)

\( (\R_+, \, \mu_{\bar{\varepsilon}}) \) space of energy levels

\( \mu_{\bar{\varepsilon}}\) on \(\R_+\)    energy law

\( ((0,1), \, \mathrm{Leb})\) space of quantiles

\( F^{\leftarrow}_{\mathbb{P}_X}\) quantile function

\( ((0,q^+), \, \mathrm{Leb}) \) space of energy quantiles

\( F^{\leftarrow}_{\mu_{\bar{\varepsilon}}}\) energy quantile function

Simulations

Computations

Modelling