On microscopic origins of generalized gradient structures

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.

Keywords: Generalized gradient structure, gradient system, evolutionary Γ-convergence, energy-dissipation principle, variational evolution, relative entropy, large-deviation principle

References in corpus (2)

• On the relation between gradient flows and the large-deviation principle, with applications to Markov chains and diffusion
• Quadratic and rate-independent limits for a large-deviations functional

Cited by in corpus (5)

• Jump processes as Generalized Gradient Flows
• Deriving GENERIC from a generalized fluctuation symmetry
• Decay to equilibrium for energy-reaction-diffusion systems
• EDP-convergence for nonlinear fast-slow reaction systems with detailed balance
• Revisit of macroscopic dynamics for some non-equilibrium chemical reactions from a Hamiltonian viewpoint