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A consistent non-linear Fokker-Planck model for a gas mixture of polyatomic molecules

Marlies Pirner

TL;DR

The paper develops a consistent, space-homogeneous Fokker-Planck model for a non-reactive two-species polyatomic gas, capturing both translational and internal-energy exchanges through intra- and interspecies collisions. Maxwellian targets $M_k$ and cross-Maxwellians $M_{kj}$ are used to drive the distributions toward equilibrium, with partial temperatures $ ext{Λ}_k$ and $ ext{Θ}_k$ that satisfy energy constraints and yield a common temperature at equilibrium. A rigorous H-theorem is established, showing the entropy $H(f_1,f_2)$ is non-increasing and equality occurs only for a universal Maxwellian with equal velocities and temperatures; positivity of all internal-energy components is ensured by suitable parameter bounds. The framework generalizes to $N$ species and is illustrated in a mono-atomic vs. polyatomic mix, where the translational and rotational exchanges simplify while preserving conservation and the H-theorem. Overall, the work provides a computationally tractable kinetic model for polyatomic gas mixtures with correct thermodynamics and tunable inter-species exchange rates.

Abstract

We consider a multi component gas mixture with translational and internal energy degrees of freedom without chemical reactions assuming that the number of particles of each species remains constant. We will illustrate the derived model in the case of two species, but the model can be generalized to multiple species. The two species are allowed to have different degrees of freedom in internal energy and are modeled by a system of kinetic Fokker-Planck equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the temperatures, H-theorem and we characterize the equilibrium as two Maxwell distributions where all temperatures coincide.

A consistent non-linear Fokker-Planck model for a gas mixture of polyatomic molecules

TL;DR

The paper develops a consistent, space-homogeneous Fokker-Planck model for a non-reactive two-species polyatomic gas, capturing both translational and internal-energy exchanges through intra- and interspecies collisions. Maxwellian targets and cross-Maxwellians are used to drive the distributions toward equilibrium, with partial temperatures and that satisfy energy constraints and yield a common temperature at equilibrium. A rigorous H-theorem is established, showing the entropy is non-increasing and equality occurs only for a universal Maxwellian with equal velocities and temperatures; positivity of all internal-energy components is ensured by suitable parameter bounds. The framework generalizes to species and is illustrated in a mono-atomic vs. polyatomic mix, where the translational and rotational exchanges simplify while preserving conservation and the H-theorem. Overall, the work provides a computationally tractable kinetic model for polyatomic gas mixtures with correct thermodynamics and tunable inter-species exchange rates.

Abstract

We consider a multi component gas mixture with translational and internal energy degrees of freedom without chemical reactions assuming that the number of particles of each species remains constant. We will illustrate the derived model in the case of two species, but the model can be generalized to multiple species. The two species are allowed to have different degrees of freedom in internal energy and are modeled by a system of kinetic Fokker-Planck equations featuring two interaction terms to account for momentum and energy transfer between the species. We prove consistency of our model: conservation properties, positivity of the temperatures, H-theorem and we characterize the equilibrium as two Maxwell distributions where all temperatures coincide.
Paper Structure (6 sections, 5 theorems, 71 equations)

This paper contains 6 sections, 5 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. The space-homogeneous Fokker-Planck approximation for a mixture of polyatomic molecules
  3. Conservation properties
  4. Positivity of the internal energies
  5. H-Theorem
  6. A mixture consisting of a mono and a polyatomic gas

Key Result

Theorem 1

Assume coll, conditions density, convexvel and veloc and assume that $\Lambda_{12}$ and $\Theta_{12}$ satisfy Then we have conservation of total energy

Theorems & Definitions (15)

  • Remark 1
  • Theorem 1: Conservation of total energy
  • proof
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2
  • proof
  • Remark 5
  • Lemma 3: Contribution to the H-theorem from the one species relaxation terms
  • ...and 5 more