Approaches for modeling polyatomic molecules in kinetic theory

University of Novi Sad

April 15, 2025

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

\text{M. Bisi, } \underline{\text{T. Borsoni}}^* \text{, M. Groppi}

Microscopic

Macroscopic

Kinetic theory: statistical description of gases

Mesoscopic

Polyatomic molecules: with internal structure

Monoatomic :

Polyatomic :

\mathrm{Ar}, \, \mathrm{He}, \dots
\mathrm{H}_2 \mathrm{O}, \, \mathrm{N}_2, \, \mathrm{CO}_2, \dots

Outline

  1. The Boltzmann equation for monoatomic gases
  2. Extension to polyatomic gases
  3. Three points of view for modeling a polyatomic molecule 

The original Boltzmann equation

[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, \cdot))(v)
v
x
+

advection

collisions

Density of molecules

 \(f \equiv f_{t}(x,v)\)

v
x
+

(monoatomic)

the collision operator

v
v_*
v'
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*}
\newcommand{\Sb}{\mathbb{S}} \sigma \in \Sb^2
\newcommand{\dd}{\mathrm{d}} \textcolor{black}{Q(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_*
\begin{align*} v+v_*&=v'+v'_* \\ |v|^2 + |v_*|^2 &= |v'|^2 + |v'_*|^2 \\ \end{align*}

Conservations

\(B(v,v_*,\sigma)\) : nature of interaction

hard spheres   

\(B(v,v_*,\sigma) = |v-v_*|\)

power law potential

\(B(v,v_*,\sigma) = b(\cos \theta)|v-v_*|^\gamma\)

...

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

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}

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) = \textcolor{blue}{\rho} \, (2 \pi \textcolor{blue}{T})^{-3/2} \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.

Illustration of the expected behaviour

\textcolor{blue}{\sqrt{T}}

Degrees of freedom and heat capacity

\Theta_{eq} = \int_{\R^3} \rho \, (2 \pi T)^{-3/2} \exp\left(-\frac{|v-u|^2}{2 T} \right) \frac12 |v- u|^2 \, \mathrm{d} v = \; \frac{3}{2}\rho T
\Theta := \int_{\R^3}f(v) \, \frac12 |v- u|^2 \, \mathrm{d} v,

Microscopic energy of \(f\):

u = \frac{1}{\rho}\int_{\R^3}f(v) \, v \, \mathrm{d} v

with

At equilibrium, \(f = M\):

\rho = \int_{\R^3}f(v) \, \mathrm{d} v,
  • Number of degrees of freedom       \(:=\)
\frac{2}{T} \times \frac{\Theta_{eq}}{\rho} = \; \; 3
  • Heat capacity at constant volume  \(:=\)
\frac{\mathrm{d}}{\mathrm{d} T} \, \frac{\Theta_{eq}}{\rho} = \; \; \frac32

monoatomic gas

  1. The Boltzmann equation for monoatomic gases
  2. Extension to polyatomic gases
  3. Three points of view for modeling a polyatomic molecule 

 polyatomic molecules have an internal structure

v
\omega
n_1
n_2
n_3
x
+

continuous

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

discrete

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

v
x
+

Density of molecules \(f \equiv f(t,x,v)\)

Density of molecules \(f \equiv f(t,x,v,\textcolor{orange}{?})\)

Internal state

? : (\omega,n_1,n_2,n_3)

Internal

energy quantile

Internal

energy level

? : \frac12\mathcal{J}|\omega|^2 + E_1 n_1 + E_2 n_2 + E_3n_3
f(t,x,v,\textcolor{orange}{\zeta})
f(t,x,v,\textcolor{gray}{q})
f(t,x,v,\textcolor{olive}{I})
f_{\textcolor{olive}{n}}(t,x,v)

[Taxman 58']

general mathematical framework

[Bisi\(\text{--}\)TB\(\text{--}\)Groppi '22]

[Gamba\(\text{--}\)Pavić-Čolić '20]

the two requirements for a polyatomic kinetic model

1. Modeling the molecule

2. Modeling the interaction between molecules

this presentation

v,\textcolor{olive}{I}
v_*,\textcolor{olive}{I_*}
v',\textcolor{olive}{I'}
v'_*,\textcolor{olive}{I'_*}

Determine the collision kernel

\(B(v,v_*,\sigma,I,I_*,I',I'_*)\)

Monchick, Mason, Hellman...

\(f(t,x,v,\textcolor{orange}{?})\)

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

general mathematical framework based on internal states

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

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

\(\displaystyle  Z(\beta) := \int_{\mathcal{E}} \exp(-\beta \, \bar{\varepsilon}(\zeta)) \, \mathrm{d} \mu(\zeta) < \infty\),         \(\forall \, \beta >0\).

1. Space of internal states:

Assumptions

measured space

Ingredients

\(\sigma\)-finiteness of \(\mu\)

measurable function

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

\(f \equiv f_{t,x}(v, \textcolor{orange}{\zeta})\) density of molecules

\textcolor{orange}{\zeta}
+
x

the polyatomic Boltzmann equation

1. internal states

v

Boltzmann equation:

v,\textcolor{orange}{\zeta}
v_*,\textcolor{orange}{\zeta_*}
v',\textcolor{orange}{\zeta'}
v'_*,\textcolor{orange}{\zeta'_*}

Collision operator:

\newcommand{\dd}{\mathrm{d}} \textcolor{black}{Q(f)(v,}\textcolor{orange}{\zeta}\textcolor{black}{) = }\iint_{\R^3 \times \mathbb{S}^2} \textcolor{orange}{\iiint_{\mathcal{E}^3}} \textcolor{black}{[f(v',\textcolor{orange}{\zeta'}) f(v'_*,\textcolor{orange}{\zeta'_*}) - f(v,\textcolor{orange}{\zeta}) f(v_*,\textcolor{orange}{\zeta_*})]} B \; \textcolor{orange}{\dd \mu^{\otimes 3}(\zeta_*,\zeta',\zeta'_*)} \, \dd v_* \, \dd \sigma
\begin{align*} v' = \frac{v+v_*}{2} + \textcolor{orange}{\sqrt{\Delta}} \; \sigma, \\ v'_* = \frac{v+v_*}{2} - \textcolor{orange}{\sqrt{\Delta}} \; \sigma,\\ \end{align*}
\newcommand{\Sb}{\mathbb{S}} \sigma \in \Sb^2,
\begin{align*} v+v_*&=v'+v'_* \\ \frac12|v|^2 + \textcolor{orange}{\varepsilon(\zeta)} + \frac12|v_*|^2 + \textcolor{orange}{\varepsilon(\zeta_*)} &= \frac12|v'|^2 + \textcolor{orange}{\varepsilon(\zeta')} + \frac12|v'_*|^2 + \textcolor{orange}{\varepsilon(\zeta'_*)} \\ \end{align*}
\begin{align*}\textcolor{orange}{\Delta} := \frac14 |v-v_*|^2 + \textcolor{orange}{\varepsilon(\zeta) + \varepsilon(\zeta_*) - \varepsilon(\zeta') - \varepsilon(\zeta'_*)} \end{align*}

Conservation of

mass, momentum, energy :

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

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

+
x

the polyatomic Boltzmann equation

2. (continuous) internal energy levels

v

Boltzmann equation:

v,\textcolor{olive}{I}
v_*,\textcolor{olive}{I_*}
v',\textcolor{olive}{I'}
v'_*,\textcolor{olive}{I'_*}

Collision operator:

\newcommand{\dd}{\mathrm{d}} \textcolor{black}{Q(f)(v,}\textcolor{olive}{I}\textcolor{black}{) = }\iint_{\R^3 \times \mathbb{S}^2} \textcolor{olive}{\iiint_{\mathbb{R}_+^3}} \textcolor{black}{[f(v',\textcolor{olive}{I'}) f(v'_*,\textcolor{olive}{I'_*}) - f(v,\textcolor{olive}{I}) f(v_*,\textcolor{olive}{I_*})]} B \; \textcolor{olive}{\varphi(I') \dd I' \, \varphi(I'_*) \dd I'_* \, \varphi(I_*) \dd I_*} \, \dd v_* \, \dd \sigma
\begin{align*} v' = \frac{v+v_*}{2} + \textcolor{olive}{\sqrt{\Delta}} \; \sigma, \\ v'_* = \frac{v+v_*}{2} - \textcolor{olive}{\sqrt{\Delta}} \; \sigma,\\ \end{align*}
\newcommand{\Sb}{\mathbb{S}} \sigma \in \Sb^2,
\begin{align*} v+v_*&=v'+v'_* \\ \frac{m}2|v|^2 + \textcolor{olive}{I} + \frac{m}2|v_*|^2 + \textcolor{olive}{I_*} &= \frac{m}2|v'|^2 + \textcolor{olive}{I'} + \frac{m}2|v'_*|^2 + \textcolor{olive}{I'_*} \\ \end{align*}
\begin{align*}\textcolor{olive}{\Delta} := \frac14 |v-v_*|^2 + \frac1m(\textcolor{olive}{I+I_* - I'-I'_*}) \end{align*}

Conservation of

mass, momentum, energy :

\implies
I

energy law

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

Entropy and equilibrium in the polyatomic case

The Boltzmann entropy:

\newcommand{\dd}{\mathrm{d}} H (f) := \textcolor{orange}{\int_{\mathcal{E}}}\int_{\R^3} (f \log f - f)(v,\textcolor{orange}{\zeta}) \, \dd v \, \textcolor{orange}{\dd \mu(\zeta)}

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

characterization of equilibria

M(v,\textcolor{orange}{\zeta}) = \rho \, \left(2 \pi T\right)^{-3/2} \, \textcolor{orange}{Z(1/T)^{-1}} \exp \left(-\frac{\frac{1}2|v-u|^2 + \textcolor{orange}{\varepsilon(\zeta)}}{T}\right)
\partial_t f_t(v,\textcolor{orange}{\zeta}) = Q(f_t)(v,\textcolor{orange}{\zeta})

Boltzmann's H Theorem

1. If \(f \equiv f_t(v,\textcolor{orange}{\zeta})\) solves

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

then

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

Degrees of freedom and heat capacity

\Theta_{eq} = \frac{3}{2}\rho T + \textcolor{orange}{\Theta_{int}(T)}
\Theta := \textcolor{orange}{\int_{\mathcal{E}}}\int_{\R^3}f(v,\textcolor{orange}{\zeta}) \left(\frac12 |v- u|^2 + \textcolor{orange}{\bar{\varepsilon}(\zeta)} \right) \mathrm{d} v \, \textcolor{orange}{\mathrm{d} \mu(\zeta)}

Microscopic energy of \(f\):

At equilibrium, \(f = M\):

  • Number of degrees of freedom      \(:=\)
\frac{2}{T} \times \frac{\Theta_{eq}}{\rho} = 3 + \textcolor{orange}{\delta(T)}
  • Heat capacity at constant volume  \(:=\)
\frac{\mathrm{d}}{\mathrm{d} T} \, \frac{\Theta_{eq}}{\rho} = \frac32 + \frac12 \frac{\mathrm{d}}{\mathrm{d}T} (T \textcolor{orange}{\delta(T)})
= 3 + \textcolor{orange}{2 \, Z(1/T)^{-1} \int_{\mathcal{E}} \bar{\varepsilon}(\zeta) \, e^{-\frac{\bar{\varepsilon}(\zeta)}{T}} \, \mathrm{d} \mu(\zeta)}
  1. The Boltzmann equation for monoatomic gases
  2. Extension to polyatomic gases
  3. Three points of view for modeling a polyatomic molecule 

parallel with probability setting leading to 3 points of view

polyatomic internal structure

probability setting

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

\( X : \Omega \to \R_+ \)   random variable

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

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

\( \bar{\varepsilon} = \varepsilon - \inf \varepsilon \)

\( (\R_+, \, \mathbb{P}_X)\) 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

\( ((0,1), \, Lebesgue) \) space of quantiles

\( F^{\leftarrow}_{\mathbb{P}_X}: (0,1) \to \R_+\)    quantile function

\( ((0,\, q^{\max}), \, Lebesgue) \) space of energy quantiles

\( F^{\leftarrow}_{\mu_{\bar{\varepsilon}}}: (0,\, q^{\max}) \to \R_+\)    energy quantile func.

\(q^{\max} := \mu(\mathcal{E})\)

\(\mathbb{P}_X = X \# \mathbb{P} \)

\(\mu_{\bar{\varepsilon}} = \bar{\varepsilon} \#\mu \)

internal   state  

internal energy level

internal energy quantile

Paradigms and a suggestion for their use         

\(\zeta \in \mathcal{E}\)

\(\mu\)

\(\bar{\varepsilon}\)

\( I \in \R_+\)

\(\mu_{\bar{\varepsilon}}\)

\( q \in (0,q^{\max})\)

\(\mathrm{Id}_{\R_+}\)

\(F_{\mu_{\bar{\varepsilon}}}^{\leftarrow}\)

Lebesgue

variable

measure

energy function

Physical modeling

& general proofs

Computations

& technical proofs

Numerical simulations

(particle-based)

well-suited for

(grounded)

Internal states

Internal energy levels

Internal energy quantiles

State-based

Energy-based

2 approaches, 3 points of view - loss of information

contains all the information

contains only the information on the energy

loss of information

A typical example for the diatomic molecule

Physical model:

  • Classical Rigid-Rotor for rotation
  • Quantum harmonic oscillator for vibration

Number of degrees of freedom (T)

Heat capacity at constant volume (T)

internal

states

internal energy levels

internal energy quantiles

\(\mathcal{E} = \R^2 \times \N\)

\(\mu = Lebesgue_{\R^2} \otimes Counting_{\N}\)

\(\varepsilon (\omega,n) =\frac12 \mathcal{J} |\omega|^2 + \left( n  + \frac12 \right) \Delta \epsilon\)

\( (\R_+,\varphi(I) dI) \)

\( \mathrm{Id}_{\R_+}\)

\( (0,+\infty) \)

\( Lebesgue_{\R^*_+} \)

\(\varphi(I) =\displaystyle \frac{2\pi}{\mathcal{J}} \left\lceil \frac{I}{\Delta \epsilon} \right\rceil \)

\(\displaystyle F^{\leftarrow}_{\mu_{\bar{\varepsilon}}}(q) = \Delta \epsilon \left(\frac{\hat{q}}{\ell(\hat{q})+1} + \frac{\ell(\hat{q})}{2} \right)\)

\(\displaystyle \hat{q} := \frac{\mathcal{J}}{2\pi \; \Delta \epsilon} \; q\),   \( \displaystyle \ell(\hat{q}) := \left\lfloor \frac{4 \, \hat{q}}{\sqrt{1 + 8 \, \hat{q}} \, + 1} \right\rfloor\)

"non-polytropic"

polyatomic models in the literature (in practice)

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

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

1.    \(\mu_{\bar{\varepsilon}}\) is a discrete measure

          supported on \(\{\epsilon_n\}_n\)

(\R_+, \, \mu_{\bar{\varepsilon}}) \, \longleftrightarrow \, (\N, \, (w_n)_n)

model with discrete energy levels

[Wang Chang-Uhlenbeck 51',

Bisi-Groppi-Spiga 2005...]

with energy levels \(\{\epsilon_n\}_n\)

2.   \(\mu_{\bar{\varepsilon}}\) has a density \(\varphi\) w.r.t.

         Lebesgue measure

model with continuous energy levels

[Borgnakke-Larsen 75',

Desvillettes 97'...]

(\R_+, \, \mathrm{d}\mu_{\bar{\varepsilon}}(I)) = (\R_+, \, \textcolor{red}{\varphi}(I) \mathrm{d}I)

States

Energy quantiles

Build

Analyse

Simulate

Energy levels

Thank you for your attention!