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On the modelling of polyatomic molecules in kinetic theory

Marzia Bisi, Thomas Borsoni, Maria Groppi

TL;DR

This paper articulates three parallel frameworks for modelling polyatomic molecular internal structure within kinetic theory: internal states, internal energy levels, and internal energy quantiles. It develops a coherent mathematical bridge among these descriptions, derives explicit forms for energy laws and quantile functions in a diatomic example, and analyzes how these descriptions propagate to mesoscopic quantities and equilibrium measures. The work highlights when state-based and energy-based descriptions are equivalent, both at equilibrium and out of equilibrium, and provides concrete Maxwellian formulations and thermodynamic quantities such as the degree of freedom count and heat capacity, including detailed diatomic expressions. The results are relevant for both analytic studies and particle-based simulations (DSMC), and they offer a flexible toolkit for handling non-polytropic gases in kinetic theory. Practically, the framework supports tailored modelling choices for complex molecular internal structures while preserving consistent links to macroscopic observables.

Abstract

This communication is both a pedagogical note for understanding polyatomic modelling in kinetic theory and a ''cheat sheet'' for a series of corresponding concepts and formulas. We explain, detail and relate three possible approaches for modelling the polyatomic internal structure, that are: the internal states approach, well suited for physical modelling and general proofs, the internal energy levels approach, useful for analytic studies and corresponding to the common models of the literature, and the internal energy quantiles approach, less known while being a powerful tool for particle-based numerical simulations such as Direct Simulation Monte-Carlo (DSMC). This note may in particular be useful in the study of non-polytropic gases.

On the modelling of polyatomic molecules in kinetic theory

TL;DR

This paper articulates three parallel frameworks for modelling polyatomic molecular internal structure within kinetic theory: internal states, internal energy levels, and internal energy quantiles. It develops a coherent mathematical bridge among these descriptions, derives explicit forms for energy laws and quantile functions in a diatomic example, and analyzes how these descriptions propagate to mesoscopic quantities and equilibrium measures. The work highlights when state-based and energy-based descriptions are equivalent, both at equilibrium and out of equilibrium, and provides concrete Maxwellian formulations and thermodynamic quantities such as the degree of freedom count and heat capacity, including detailed diatomic expressions. The results are relevant for both analytic studies and particle-based simulations (DSMC), and they offer a flexible toolkit for handling non-polytropic gases in kinetic theory. Practically, the framework supports tailored modelling choices for complex molecular internal structures while preserving consistent links to macroscopic observables.

Abstract

This communication is both a pedagogical note for understanding polyatomic modelling in kinetic theory and a ''cheat sheet'' for a series of corresponding concepts and formulas. We explain, detail and relate three possible approaches for modelling the polyatomic internal structure, that are: the internal states approach, well suited for physical modelling and general proofs, the internal energy levels approach, useful for analytic studies and corresponding to the common models of the literature, and the internal energy quantiles approach, less known while being a powerful tool for particle-based numerical simulations such as Direct Simulation Monte-Carlo (DSMC). This note may in particular be useful in the study of non-polytropic gases.

Paper Structure

This paper contains 27 sections, 34 equations, 3 figures, 4 tables.

Table of Contents

  1. Modelling a polyatomic molecule
  2. A general description based on internal states
  3. Descriptions based on internal energy
  4. Energy levels
  5. Energy quantiles
  6. Links between descriptions
  7. Summary and use of the three descriptions
  8. Model combination
  9. Independent combination of two state-based models
  10. Separated- and total-energy-based corresponding descriptions
  11. Separated-internal energy levels
  12. Separated-internal energy quantiles
  13. Link between separated- and total-internal energy levels descriptions
  14. A typical model for a diatomic molecule
  15. Molecular model and internal states description
  16. ...and 12 more sections

Figures (3)

  • Figure 1: Schematic representation of the three main points of view for describing the internal structure of a polyatomic molecule.
  • Figure 2: Plots of energy laws and quantile functions associated with the discussed model.
  • Figure 3: Plot of the total number of degrees of freedom and heat capacity at constant volume associated with the model described in Sec. \ref{['section2']}.