Nonlinear Vlasov-Fokker-Planck equations: From generalized Wasserstein gradient flow to GENERIC structure

TL;DR

This paper explores the GENERIC formulation of nonlinear Vlasov-Fokker-Planck equations, revealing a trajectory-based gradient flow perspective that incorporates entropy, energy, and a novel metric structure.

Contribution

It introduces a trajectory-based gradient flow framework for the nonlinear Vlasov-Fokker-Planck equation within the GENERIC formalism, highlighting the influence of nonlinear terms.

Findings

Recasting the evolution as a generalized Wasserstein gradient flow.

Derivation of a partial HWI inequality from metric space decomposition.

Identification of the impact of nonlinear terms on free energy dissipation.

Abstract

We study the GENERIC (General Equation for Non-Equilibrium Reversible Irreversible Coupling) formulation of the nonlinear Vlasov-Fokker-Planck equation from the perspective of gradient flows along trajectories. After pulling back the reversible component, the evolution can be recast as a generalized Wasserstein gradient flow. The associated free energy functional consists of an entropy term and a conservative energy term, while the metric is induced by the Onsager operator. This trajectory based viewpoint shows that the trajectorial rate of free energy dissipation captures the influence of the nonlinear term, an effect that is not directly apparent at the macroscopic level. Finally, partial degeneracy yields a decomposition of the underlying metric space, which in turn enables the derivation of a partial HWI inequality.