A non-local quasi-equilibrium state in the Bhatnagar-Gross-Krook Boltzmann equation for thermo-hydrodynamics: Conservation laws, the Boltzmann H-theorem, and the fluctuation-dissipation theorem

TL;DR

This paper introduces a non-local quasi-equilibrium state in the BGK Boltzmann equation, enhancing the modeling of thermo-hydrodynamics by explicitly incorporating longer-range interactions and ensuring consistency with fundamental physical principles.

Contribution

It develops a non-local formulation of the equilibrium state in the BGK model, allowing flexible transport coefficients and equation of state while maintaining key physical laws.

Findings

Ensures conservation laws are satisfied in the non-local model.

Maintains the Boltzmann H-theorem with the new formulation.

Aligns with the fluctuation-dissipation theorem.

Abstract

The Bhatnagar-Gross-Krook (BGK) Boltzmann equation with the Maxwellian-Boltzmann-type equilibrium state leads to the set of thermo-hydrodynamic equations such as the continuity, the Navier-Stokes, and the heat-transfer equations in the scaling limit. With its efficient and promising framework handling multi-scale physics, the collision model has been studied with both of theoretical and numerical approaches to apply it for extensive flow conditions such as the flexible choices of the Prandtl number. In this study, using an analytic technique of the kinetic generator, we employ a non-local formulation for the equilibrium state leading to the thermo-hydrodynamic equations with flexible choices of transport coefficients and the equation of state (EOS). The equilibrium state includes the quasi-equilibrium state intrinsically, being formulated with the non-local macroscopic quantities so that the longer-range interaction is explicitly involved. According to the new formulation, the consistency with conservation laws, the Boltzmann H-theorem, and the fluctuation-dissipation theorem are examined.