Eigenfunctions for Liouville Operators, Classical Collision Operators, and Collision Bracket Integrals in Kinetic Theory

TL;DR

This paper develops new forms for collision bracket integrals in dense fluid kinetic theory using eigenfunctions of the Liouville operator, making numerical computation more feasible.

Contribution

It introduces alternative forms of collision integrals based on Liouville eigenfunctions, simplifying numerical simulations in dense fluid kinetic theory.

Findings

Derived forms resemble time correlation functions.

Simplified collision integrals are more suitable for numerical methods.

Connects dense fluid theory with dilute gas approximations.

Abstract

In the kinetic theory of dense fluids the many-particle collision bracket integral is given in terms of a classical collision operator defined in the phase space. To find an algorithm to compute the collision bracket integrals, we revisit the eigenvalue problem of the Liouville operator and re-examine the method previously reported[Chem. Phys. 20, 93(1977)]. Then we apply the notion and concept of the eigenfunctions of the Liouville operator and knowledge acquired in the study of the eigenfunctions to obtain alternative forms for collision integrals. One of the alternative forms is given in the form of time correlation function. This form, on an additional approximation, assumes a form reminiscent of the Chapman-Enskog collision bracket integral for dilute gases. It indeed gives rise to the latter in the case of two particles. The alternative forms obtained are more readily amenable to numerical simulation methods than the collision bracket integras expressed in terms of a classical collision operator, which requires solution of classical Lippmann-Schwinger integral equations. This way, the aforementioned kinetic theory of dense fluids is made more accessible by numerical computation/simulation methods than before.