Quasi-Equilibrium Closure Hierarchies for The Boltzmann Equation

TL;DR

This paper introduces the Triangle Entropy Method for constructing explicit closures in kinetic equations, allowing for detailed treatment of scattering rates and leading to improved macroscopic models with renormalized transport coefficients.

Contribution

The paper develops a new explicit closure method for kinetic equations that incorporates scattering rates as independent variables, enhancing the modeling of nonequilibrium processes.

Findings

Closure of hydrodynamic chains is achieved explicitly.

Scattering processes lead to renormalized transport coefficients.

Method applies to any kinetic equation with a thermodynamic Lyapunov function.

Abstract

In this paper, explicit method of constructing approximations (the Triangle Entropy Method) is developed for nonequilibrium problems. This method enables one to treat any complicated nonlinear functionals that fit best the physics of a problem (such as, for example, rates of processes) as new independent variables. The work of the method was demonstrated on the Boltzmann's--type kinetics. New macroscopic variables are introduced (moments of the Boltzmann collision integral, or scattering rates). They are treated as independent variables rather than as infinite moment series. This approach gives the complete account of rates of scattering processes. Transport equations for scattering rates are obtained (the second hydrodynamic chain), similar to the usual moment chain (the first hydrodynamic chain). Various examples of the closure of the first, of the second, and of the mixed hydrodynamic chains are considered for the hard spheres model. It is shown, in particular, that the complete account of scattering processes leads to a renormalization of transport coefficients. The method gives the explicit solution for the closure problem, provides thermodynamic properties of reduced models, and can be applied to any kinetic equation with a thermodynamic Lyapunov function.