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

x
+
+

 polyatomic molecules have an internal structure

v
?
x
+
v
x

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

v
x
+
I

\(\bullet\) 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
+
x

the polyatomic Boltzmann equation

v

Boltzmann equation:

I

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

v

Space homogeneous Boltzmann equation

I

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
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

(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

If

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

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,}

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

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

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,}

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

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

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,}

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

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.

If

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)

(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

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!

Presentation IRMAR

By Thomas Borsoni

Presentation IRMAR

  • 2

More from Thomas Borsoni