On microscopic origins of generalized gradient structures

TL;DR

This paper explores the microscopic origins of generalized gradient structures, linking them to large-deviation principles and a new convergence method called EDP-convergence, with applications to diffusion and membrane models.

Contribution

It introduces two natural origins for generalized gradient structures: large-deviation principles for jump processes and EDP-convergence for deriving new structures from classical systems.

Findings

Poissonian jump processes lead to cosh-type dissipation potentials.

EDP-convergence can produce generalized gradient systems from classical ones.

Applications include membrane models and reaction-diffusion systems.

Abstract

Classical gradient systems have a linear relation between rates and driving forces. In generalized gradient systems we allow for arbitrary relations derived from general non-quadratic dissipation potentials. This paper describes two natural origins for these structures. A first microscopic origin of generalized gradient structures is given by the theory of large-deviation principles. While Markovian diffusion processes lead to classical gradient structures, Poissonian jump processes give rise to cosh-type dissipation potentials. A second origin arises via a new form of convergence, that we call EDP-convergence. Even when starting with classical gradient systems, where the dissipation potential is a quadratic functional of the rate, we may obtain a generalized gradient system in the evolutionary $\Gamma$-limit. As examples we treat (i) the limit of a diffusion equation having a thin layer of low diffusivity, which leads to a membrane model, and (ii) the limit of diffusion over a high barrier, which gives a reaction-diffusion system.