Transport and Equilibrium in Non-Conservative Systems

TL;DR

This paper introduces a grand canonical framework for a non-conservative McKean-Vlasov system, demonstrating gradient flow structure, developing a JKO-scheme, and showing exponential convergence to equilibrium independent of volume.

Contribution

It formulates a grand canonical version of the McKean-Vlasov equation, establishes a gradient flow structure, and proves volume-independent exponential convergence using a novel JKO-scheme.

Findings

Convergence to uniform state is exponential and volume-independent.

Developed a JKO-scheme applicable to non-conservative problems.

Grand canonical approach offers advantages over conservative systems.

Abstract

We study, in finite volume, a grand canonical version of the McKean-Vlasov equation where the total particle content is allowed to vary. The dynamics is anticipated to minimize an appropriate grand canonical free energy; we make this notion precise by introducing a metric on a set of positive Borel measures without pre-prescribed mass and demonstrating that the dynamics is a gradient flow with respect to this metric. Moreover, we develop a JKO-scheme suitable for these problems. The latter ideas have general applicability to a class of second order non-conservative problems. For this particular system we prove, using the JKO-scheme, that (under certain assumptions) convergence to the uniform stationary state is exponential with a rate which is independent of the volume. By contrast, in related conservative systems, decay rates scale - at best - with the square of the characteristic length of the system. This suggests that a grand canonical approach may be useful for both theoretical and computational study of large scale systems.