Geometric dissipation in kinetic equations

TL;DR

This paper introduces a symplectic variational method for modeling dissipation in kinetic equations, resulting in a double bracket structure that preserves certain invariants like entropy, with applications to Vlasov equations.

Contribution

It develops a novel symplectic variational framework that models dissipation while maintaining key invariants in kinetic equations.

Findings

The approach yields a double bracket structure in phase space.

Vlasov equations admit measure-valued single-particle solutions.

Total entropy remains conserved as a Casimir.

Abstract

A new symplectic variational approach is developed for modeling dissipation in kinetic equations. This approach yields a double bracket structure in phase space which generates kinetic equations representing coadjoint motion under canonical transformations. The Vlasov example admits measure-valued single-particle solutions. Such solutions are reversible; and the total entropy is a Casimir, and thus is preserved.