Presentation IRMAR

Quasi-resonant collisions in kinetic theory and bi-temperature systems

Séminaire d'analyse numérique

IRMAR

November 27, 2025

\(\phantom{x}^*\)post-doc at CERMICS, École des Nationale des Ponts et Chaussées, supervised by Virginie Ehrlacher and Tony Lelièvre

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

a peculiar polyatomic Boltzmann model

Microscopic

Macroscopic

Kinetic theory: statistical description of gases

Mesoscopic

gaz large collection of molecules

Part 1. Standard polyatomic Boltzmann

Part 2. Polyatomic Boltzmann with quasi-resonant collisions

Outline

How to model quasi-resonance
Peculiar characteristic of the dynamic
Numerical experiments

Part 1. Standard polyatomic Boltzmann

polyatomic molecules have an internal structure

position, velocity

position, velocity, angular velocity, vibration modes,...

monoatomic

polyatomic

Internal state

\textcolor{orange}{?} : (\omega,n_1,n_2,n_3)

Internal

energy quantile

Internal

energy level

\textcolor{olive}{?} : I =\frac12\mathcal{J}|\omega|^2 + E_1 n_1 + E_2 n_2 + E_3n_3

x,v

x,v, \textcolor{orange}{?}

How to describe the internal structure?

[Borgnakke\(\text{--}\)Larsen 75', Desvillettes 97']

[Wang-Chang\(\text{--}\)Uhlenbeck 51', Waldmann 57', Snider 60']

[Taxman 58', B.\(\text{--}\)Bisi\(\text{--}\)Groppi '23]

\omega

n_1

n_2

n_3

I \in \R_+

q \in \R_+

Description of polyatomic molecules we choose

\(\bullet\) position, velocity and internal energy level

x \in \R^3, \; v \in \R^3, \; \textcolor{olive}{I \in \R_+}

We consider here a molecular description by

\(\bullet\) energy carried by the molecule:

\frac{1}{2} |v|^2 + \textcolor{olive}{I}

kinetic energy

internal energy

\(\bullet\) internal energy law of the molecule

\textcolor{olive}{\varphi : I \in \R_+ \; \mapsto \; \varphi (I) \in \R_+}

describes how the energy levels are distributed

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

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

the polyatomic Boltzmann equation

Boltzmann equation:

energy of the molecule

= \frac12|v|^2 + I

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

Description of a gaz as an infinite collection of molecules

Each molecule either moves in straight line of collides with another molecule

\partial_t f_{t}(v,I) = Q(f_{t})(v,I)

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

the space homogeneous

polyatomic Boltzmann equation

Space homogeneous Boltzmann equation

energy of the molecule

= \frac12|v|^2 + I

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

Assume that the gaz is homogeneous in space

Dynamic driven by collisions between molecules

v \cdot \nabla_x

the collision operator \(Q\)

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

(momentum)

(total energy)

energy of the molecule

= \frac12|v|^2 + I

Collision operator \(Q\)

\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{purple}{B} \; \varphi(I_*)\dd I_* \varphi(I')\dd I' \varphi(I'_*)\dd 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*}

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

Collision kernel

During a collision between two molecules:

parametrization

internal energy law

\partial_t f_{t}(v,I) = Q(f_{t})(v,I),

Entropy and equilibrium

Boltzmann entropy functional

\mathcal{M}(v,I) = \textcolor{blue}{\rho}\, Z(\textcolor{blue}{T})^{-1} \, \exp \left(-\frac{|v-\textcolor{blue}{u}|^2}{2 \, \textcolor{blue}{T}} - \frac{I}{\textcolor{blue}{T}} \right)

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

Boltzmann's H-theorem

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

(ii) Equilibrium

then

\displaystyle \frac{\mathrm{d} H(f_t)}{\mathrm{d} t} =: -D(f_t) \leqslant 0.

entropy dissipation functional

For any

g \equiv g(v,I),

D(g) = 0 \quad \iff \quad g = \mathcal{M}

With \(\mathcal{M}\) the Maxwellian (Gibbs) distribution

\(\textcolor{blue}{\rho}\) average density

\(\textcolor{blue}{u}\) bulk velocity

\(\textcolor{blue}{T}\) temperature

Part 2. Polyatomic Boltzmann with quasi-resonant collisions

quasi-resonant collisions

v,I

v_*,I_*

v',I'

v'_*,I'_*

(separation kinetic/internal)

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

v,I

v_*,I_*

v',I'

v'_*,I'_*

I+I_* \approx I' + 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*}

quasi-resonant possible collisions

How to make it rigorous?

(almost separation kinetic/internal)

resonant possible collisions

I+I_* = I' + 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*}

standard possible collisions

Part 2. Polyatomic Boltzmann with quasi-resonant collisions

How to model quasi-resonance

quasi-resonant collisions by truncation of the kernel

Set of possible collisions \(\equiv\) support of the collision kernel

resonant possible collisions

v,I

v_*,I_*

v',I'

v'_*,I'_*

I+I_* = I' + 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*}

B^{\rm \textcolor{grey}{std}}

Let

a standard collision kernel

\mathrm{supp}(B^{\rm \textcolor{grey}{std}})

standard possible collisions

\mathrm{supp}(B^{\rm {res}})

``B^{\textcolor{green}{\text{res}}} = B^{\rm \textcolor{grey}{std}} \times \textcolor{green}{\delta_{I+I_* = I'+I'_*}}"

quasi-resonant collisions by truncation of the kernel

Set of possible collisions \(\equiv\) support of the collision kernel

quasi-resonant possible collisions

v,I

v_*,I_*

v',I'

v'_*,I'_*

I+I_* \approx_\varepsilon I' + 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*}

B^{\rm \textcolor{grey}{std}}

Let

a standard collision kernel

\mathrm{supp}(B^{\rm \textcolor{grey}{std}})

\varepsilon

B^{\textcolor{red}{\text{q-res}}}_{\textcolor{red}{\varepsilon}} = B^{\rm \textcolor{grey}{std}} \times \textcolor{red}{\frac{c \,\mathbf{ \bm{1}_{\mathcal{V}_{\varepsilon}}}}{\textcolor{red}{\varepsilon}}}

\textcolor{black}{\forall} \varepsilon \textcolor{black}{> 0,}

\(\mathcal{V}_{\varepsilon}\)

\mathrm{supp}(B^{\text{\textcolor{red}{q-res}}}_{\textcolor{red}{\varepsilon}})

Family of collision kernels:

truncation family

the truncation family

B^{\rm \textcolor{grey}{std}}

a standard collision kernel

\textcolor{black}{\forall} \varepsilon \textcolor{black}{> 0,}

\(\mathcal{V}_{\varepsilon}\)

The collision is in \(\textcolor{red}{\mathcal{V}_{\varepsilon}}\) if

\left| \eta(R) - \eta(R') \right| \leqslant \textcolor{red}{\varepsilon}

Pre-and post-collision energy ratios:

\in [0,1]

A diffeomorphism \(\eta : (0,1) \to \R\)

Remark:

R = R'

\iff

the collision is resonant

The collision is in \(\textcolor{green}{\mathcal{V}_{0}}\) iff

\(\mathcal{V}_{0}\)

R' =

R =

pre-collision kinetic energy

total energy

post-collision kinetic energy

total energy

the truncation family

\textcolor{black}{\forall} \varepsilon \textcolor{black}{> 0,}

\(\mathcal{V}_{\varepsilon}\)

The collision is in \(\textcolor{red}{\mathcal{V}_{\varepsilon}}\) if

\left| \log \frac{\frac{R}{R'}}{\frac{1-R}{1-R'}} \right| \leqslant \textcolor{red}{\varepsilon}

Pre-and post-collision energy ratios:

R' =

R =

\in [0,1]

pre-collision kinetic energy

total energy

Take \(\eta : R \in (0,1) \mapsto \log \frac{R}{1-R}\)

In practice:

controls a ratio of ratios of ratios!

B^{\rm \textcolor{grey}{std}}

a standard collision kernel

Remark:

R = R'

\iff

the collision is resonant

The collision is in \(\textcolor{green}{\mathcal{V}_{0}}\) iff

post-collision kinetic energy

total energy

\newcommand{\dd}{\mathrm{d}} Q^{\textcolor{red}{\text{q-res}}}_{\textcolor{red}{\varepsilon}}(f)(v,I) = \iint_{\R^3 \times \mathbb{S}^2} \iiint_{(\R_+)^3} \big[ f' f'_* - f f_* \big] \times B^{\textcolor{red}{\text{q-res}}}_{\textcolor{red}{\varepsilon}} \times \varphi(I'_*) \, \dd I'_* \, \varphi(I') \, \dd I' \, \varphi(I_*) \, \dd I_* \, \dd v_* \, \dd \sigma,

Family of collision kernels:

Family of collision operators:

\textcolor{black}{\forall} \varepsilon \textcolor{black}{> 0,}

The quasi-resonant Boltzmann dynamics

\textcolor{black}{\forall} \varepsilon \textcolor{black}{> 0,}

Family of quasi-resonant Boltzmann dynamics

\textcolor{black}{\forall} \varepsilon \textcolor{black}{> 0,}

\partial_t f^{\textcolor{red}{\varepsilon}}_{t} = Q_{\textcolor{red}{\varepsilon}}^{\textcolor{red}{\text{q-res}}}(f^{\textcolor{red}{\varepsilon}}_{t})

RESONANT ASYMPTOTICS \(\varepsilon \to 0\)

Resonant asymptotics

Q^{\textcolor{red}{\text{q-res}}}_{\textcolor{red}{\varepsilon}}(f) \underset{\textcolor{red}{\varepsilon} \to 0}{\longrightarrow} Q^{\rm \textcolor{green}{res}} (f),

Q^{\rm \textcolor{green}{res}} (f)(v,I) = \iint_{\R^3 \times \mathbb{S}^2} \iint_{(\R_+)^{\textcolor{green}{2}}} \big[ f' f'_* - f f_* \big] \times B^{\rm \textcolor{green}{res}} \times \varphi(I+I_* - I') \, E \,\varphi(I') \, \mathrm{d} I' \, \varphi(I_*) \, \mathrm{d} I_* \, \mathrm{d} v_* \, \mathrm{d} \sigma,

``B^{\textcolor{green}{\text{res}}} = B^{\rm \textcolor{grey}{std}} \times \textcolor{green}{\delta_{I+I_* = I'+I'_*}}"

\textcolor{black}{\forall} \varepsilon \textcolor{black}{> 0,}

we expect the quasi-resonant dynamic to be "close" to the resonant one

Part 2. Polyatomic Boltzmann with quasi-resonant collisions

Peculiar characteristic of the dynamic

Resonant and Quasi-resonant long-time properties

and consequent expected behaviour

\mathcal{M}_2(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

kinetic

temperature

internal

temperature

\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

\partial_t f^{\textcolor{red}{\varepsilon}}_{t} = Q_{\textcolor{red}{\varepsilon}}^{\textcolor{red}{\text{q-res}}}(f^{\textcolor{red}{\varepsilon}}_{t})

\partial_t f^{\textcolor{green}{0}}_{t} = Q^{\textcolor{green}{\text{res}}}(f^{\textcolor{green}{0}}_{t})

For the resonant dynamic

For the quasi-resonant dynamic

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)

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

(1)

(2)

Expected behaviour of the quasi-resonant dynamics 1/2

\partial_t f^{\textcolor{red}{\varepsilon}}_{t} = Q_{\textcolor{red}{\varepsilon}}^{\textcolor{red}{\text{q-res}}}(f^{\textcolor{red}{\varepsilon}}_{t})

If \(\varepsilon \ll 1\), we expect

Explicit ODE system on the two temperatures?

Explicit Landau-Teller ODE system

\partial_t f^{\textcolor{red}{\varepsilon}}_{t} = Q_{\textcolor{red}{\varepsilon}}^{\textcolor{red}{\text{q-res}}}(f^{\textcolor{red}{\varepsilon}}_{t})

Proposition.

with

f^{\textcolor{red}{\varepsilon}}_{t=0} = \mathcal{M}_2 \propto \exp\left(-\frac{|v-\textcolor{blue}{u}|^2}{2 \, \textcolor{blue}{T^0_k}} - \frac{I}{\textcolor{blue}{T^0_i}} \right),

\displaystyle\frac{\mathrm{d} T_i(t)}{\mathrm{d} t}_{|t=0} = \varepsilon^2 \, \alpha(T_i^0, T_k^0) \; (T_k^0 - T_i^0) + o(\varepsilon^2)

then for a certain family of kernels \(B^{\textcolor{grey}{std}}\) and energy laws \(\varphi\), we have

with \(\alpha\) explicit.

Internal temperature at time \(t\):

T_i(t) \leftrightarrow \iint_{\R^3 \times \R_+} \, I \, f_t^{\textcolor{red}{\varepsilon}}(v,I) \, \mathrm{d} v \, \varphi(I)\, \mathrm{d} I

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)

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 \, \alpha[\overline{T}_k, \overline{T}_i] (\overline{T}_i - \overline{T}_k),\\ &\overline{T}_i'(t) = \e^2 \; \widetilde{\alpha}[\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)

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

expected behaviour of the QUASI-RESONANT dynamics 2/2

let's check this numerically!

Part 2. Polyatomic Boltzmann with quasi-resonant collisions

Numerical experiment

Goal. Check if

(T_i(t), \, T_k(t)) \approx (\overline{T}_i(t), \, \overline{T}_k(t))

temperatures associated with the quasi-resonant Boltzmann dynamics

temperatures of the Landau-Teller ODE system

Simulation of the quasi-resonant Boltzmann equation with DSMC

\(N\) numerical particles, each with velocity \(v\) and energy quantile \(q\)

at each time step, randomly select couples of particles to collide, according to the collision kernel

for each collision, randomly draw the new states of the particles, according to the collision kernel

Numerical experiment

\bullet \; T_k^0 = 1, \quad T_i^0 = 50, \quad T_{eq} = 20.6,

(T_i(t), \, T_k(t)) \approx (\overline{T}_i(t), \, \overline{T}_k(t))

Parameters

\varepsilon = 10^{-1},

\varphi(I) = 1,

\textcolor{black}{\bullet \; B^{\text{q-res}}_\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.

Conclusion

The distribution stays a two-temperature Maxwellian at all times

For the quasi-resonant Boltzmann dynamic

The two temperatures are related to a Landau-Teller ODE system

This ODE system is explicit from the Boltzmann model in some cases

Thank you for your attention!