Quasi-resonant collisions:

a kinetic setting for bi-temperature modeling

WASCOM

June 12, 2025

\(\phantom{x}^*\)CERMICS, École des Ponts, France

\underline{\text{T. Borsoni}}^*, \text{ L. Boudin, J. Mathiaud, F. Salvarani}

a peculiar polyatomic Boltzmann model

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

\(f \equiv f_{t,x}(v, I)\) density of molecules

+
x

the polyatomic Boltzmann equation

with internal energy levels description

v

Boltzmann equation:

v,I
v_*,I_*
v',I'
v'_*,I'_*
\begin{align*} v+v_*&=v'+v'_* \\ \frac12|v|^2 + I + \frac12|v_*|^2 + I_* &= \frac12|v'|^2 + I' + \frac12|v'_*|^2 + I'_* \\ \end{align*}

Conservation laws

I

Collision operator:

\newcommand{\dd}{\mathrm{d}} Q(f)(v,I) = \iint_{\R^3 \times \mathbb{S}^2} \iiint_{(\R_+)^3} [f(v',I') f(v'_*,I'_*) - f(v,I) f(v_*,I_*)] \; \textcolor{green}{B} \; \dd \mu^{\otimes 3}(I_*,I',I'_*) \, \dd v_* \, \dd \sigma
\begin{align*} v' = \frac{v+v_*}{2} + \sqrt{\Delta} \; \sigma, \\ v'_* = \frac{v+v_*}{2} - \sqrt{\Delta} \; \sigma,\\ \end{align*}
\newcommand{\Sb}{\mathbb{S}} \sigma \in \Sb^2,
\begin{align*}\Delta := \frac14 |v-v_*|^2 + I + I_* - I' - I'_* \end{align*}
\implies

(momentum)

(total energy)

     \(\textcolor{green}{B \equiv B(v,v_*, \sigma,I,I_*, I',I'_*)} > 0 \iff \) the collision is possible \(\iff \Delta \geq 0\) 

Collision kernel:

total energy

= \frac12|v|^2 + I
\partial_t f_{t}(v,I) = Q(f_{t})(v,I)

\(f \equiv f_{t}(v, I)\) density of molecules

the space-homogeneous

polyatomic Boltzmann equation

Boltzmann equation:

v,I
v_*,I_*
v',I'
v'_*,I'_*
\mathcal{M}(v,I) \propto\, \exp \left(-\frac{|v-\textcolor{blue}{u}|^2}{2 \, \textcolor{blue}{T}} - \frac{I}{\textcolor{blue}{T}} \right)

Equilibrium distribution :

(momentum)

(total energy)

+
x
v
I

total energy

= \frac12|v|^2 + I
\begin{align*} v+v_*&=v'+v'_* \\ \frac12|v|^2 + I + \frac12|v_*|^2 + I_* &= \frac12|v'|^2 + I' + \frac12|v'_*|^2 + I'_* \\ \end{align*}

Conservation laws

quasi-resonant collisions

\(\bullet\) What and why

\(\bullet\) How

\(\bullet\) Characteristics

Conservation laws

(momentum)

(internal energy)

(kinetic energy)

(total energy)

separately

kinetic + internal

exactly

resonant

collision

approximately

quasi-resonant

collision

Observed experimentally, e.g. \(\mathrm{CO}_2\)

Polyatomic Boltzmann model with a collision kernel with restricted support

to select only quasi-resonant collisions

  1. ​Boltzmann model where at "all times'' the solution is close to a two-temperature Maxwellian
  2. Derivation of an ODE system on the two temperatures: Landau-Teller relaxation system

explicit computations

unique feature of quasi-resonant

resonant

possible collisions

The model of resonant collisions as singular restriction of the support of the collision kernel \(B\)

\mathcal{M}(v,I) \propto\, \exp \left(-\frac{|v-\textcolor{blue}{u}|^2}{2 \, \textcolor{blue}{T_k}} - \frac{I}{\textcolor{blue}{T_i}} \right)

Equilibrium:

two distinct temperatures

v,I
v_*,I_*
v',I'
v'_*,I'_*
\begin{align*} I+I_* &= I' + I'_*, \end{align*}
\begin{align*} v+v_*&=v'+v'_*, \\ \frac12|v|^2 + I + \frac12|v_*|^2 + I_* &=\frac12|v'|^2 + I' + \frac12|v'_*|^2 + I'_*, \end{align*}

(separation kinetic/internal)

kinetic

temperature

internal

temperature

\(\equiv\) support of the collision kernel \(B\)

\varepsilon

Our model for quasi-resonant collisions as tight restriction of the support OF THE COLLISION KERNEL \(B\)

v,I
v_*,I_*
v',I'
v'_*,I'_*
\mathcal{M}(v,I) \propto\, \exp \left(-\frac{|v-\textcolor{blue}{u}|^2}{2 \, \textcolor{blue}{T}} - \frac{I}{\textcolor{blue}{T}} \right)

Equilibrium:

two distinct temperatures

same

temperature

one single temperature

\begin{align*} I+I_* &\approx I' + I'_*, \end{align*}
\begin{align*} v+v_*&=v'+v'_*, \\ \frac12|v|^2 + I + \frac12|v_*|^2 + I_* &=\frac12|v'|^2 + I' + \frac12|v'_*|^2 + I'_*, \end{align*}

quasi-resonant

possible collisions

\(\equiv\) support of the collision kernel \(B\)

Expected behaviour of the quasi-resonant dynamics

time

short time

long time

relaxation towards a

two-temperature 

Maxwellian

\propto \exp \left(-\frac{|v-\textcolor{blue}{u}|^2}{2 \, \textcolor{blue}{T_k}} - \frac{I}{\textcolor{blue}{T_i}} \right)
\partial_t f_{t} = Q_\varepsilon(f_{t})
  • the solution (almost) remains of two-temperature Mawellian shape
  • the kinetic (\(\textcolor{blue}{T_k}\)) and internal  (\(\textcolor{blue}{T_i}\)) temperatures relax towards each other, (almost) following Landau-Teller ODE system \((LT)\)
(T_i(t), \, T_k(t)) \approx (\overline{T}_i(t), \, \overline{T}_k(t))
\newcommand{\dd}{\mathrm{d}} \newcommand{\e}{\varepsilon} \begin{equation*} \begin{cases} &\overline{T}_k'(t) = \e^2 \, P[\overline{T}_k, \overline{T}_i] (\overline{T}_i - \overline{T}_k),\\ &\overline{T}_i'(t) = \e^2 \; \widetilde{P}[\overline{T}_k, \overline{T}_i] (\overline{T}_k - \overline{T}_i), \end{cases} \quad {\small(\overline{T}_i(0),\overline{T}_k(0)) = (T_i(0),T_k(0))}. \end{equation*}
(LT)

(1)

(2)

(2)

Long-time behaviour: Landau-Teller relaxation of  (\(\textcolor{blue}{T_k}\)) and (\(\textcolor{blue}{T_i}\)) towards each other

Numerical experiment

\bullet \; T_k^0 = 1, \quad T_i^0 = 50, \quad T_{eq} = 20.6,
  • Simulation of quasi-resonant Boltzmann with DSMC
  • Solving of Landau-Teller ODE system
\partial_t f_{t} = Q_\varepsilon(f_{t})
\newcommand{\dd}{\mathrm{d}} \newcommand{\e}{\varepsilon} \begin{equation*} \begin{cases} &\overline{T}_k'(t) = \e^2 \; P[\overline{T}_k, \overline{T}_i] (\overline{T}_i - \overline{T}_k),\\ &\overline{T}_i'(t) = \e^2 \; \widetilde{P}[\overline{T}_k, \overline{T}_i] (\overline{T}_k - \overline{T}_i), \end{cases} \quad {\small(\overline{T}_i(0),\overline{T}_k(0)) = (T_i(0),T_k(0))}. \end{equation*}
  • Comparison of \(T_i\) and \(\overline{T}_i\) to check if indeed
(T_i(t), \, T_k(t)) \approx (\overline{T}_i(t), \, \overline{T}_k(t))

get \(T_k\) and \(T_i\)

Parameters

\varepsilon = 10^{-1},
\mathrm{d} \mu(I) = \mathrm{d} I,
\textcolor{black}{\bullet \; B_\varepsilon(v,v_*,\sigma,I,I_*,I',I'_*) = \frac{\mathbf{1}_{[I'+I'_* \leqslant E]}}{2 \pi} \times \frac{E^3}{E_i \, E_i' \, \sqrt{E_k}} \times } \frac{\mathbf{1}_{[|\log(E_k/E_i) - \log(E_k' / E_i')| \leqslant \varepsilon]}}{\varepsilon}\textcolor{black}{,}
E_k = \frac14|v-v_*|^2, \quad E_i = I+I_*, \quad E_k' = \frac14|v'-v'_*|^2, \quad E'_i = I'+I'_*, \quad E = E_k + E_i = E_k'+ E_i'.
N_{\mathrm{DSMC}} = 10^5.

Thank you for your attention!

TB, Boudin, Mathiaud, Salvarani: A kinetic model for polyatomic gas with quasi-resonant collisions leading to bi-temperature relaxation processes, preprint (2025).

  • Rigorous proof that \(f_t\) close to two-temperature maxwellian at all times
  • Obtention of Landau-Teller-type equation for general kernels

Perspectives