Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line

TL;DR

This paper introduces a reversible particle process model on the real line that captures colloidal fluid motion and provides a rigorous mathematical framework for a corrected Dean-Kawasaki equation, highlighting a new class of Wasserstein diffusions.

Contribution

It presents a novel family of reversible coalescing-fragmentating processes with local interactions and introduces new equilibrium measures and Dirichlet forms for these dynamics.

Findings

Provides a measure-valued process solving a corrected Dean-Kawasaki equation.

Identifies the intrinsic metric as the quadratic Wasserstein distance.

Establishes a new example of Wasserstein diffusion.

Abstract

We introduce a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction on the real line as mathematically rigorous description of colloidal motion of fluids. The associated measure-valued process provides a weak solution to a corrected Dean-Kawasaki equation for supercooled liquids without dissipation. Our construction is based on the introduction and analysis of a fundamentally new family of equilibrium measures for the associated dynamics and their Dirichlet forms. We identify the intrinsic metric as the quadratic Wasserstein distance, which makes the process a non-trivial example of Wasserstein diffusion.