WIAS Preprint No. 2148, (2015)

On microscopic origins of generalized gradient structures



Authors

  • Liero, Matthias
    ORCID: 0000-0002-0963-2915
  • Mielke, Alexander
    ORCID: 0000-0002-4583-3888
  • Peletier, Mark A.
  • Renger, D. R. Michiel
    ORCID: 0000-0003-3557-3485

2010 Mathematics Subject Classification

  • 35K55 35Q82 49S05 49J40 49J45 60F10 60J25

Keywords

  • Generalized gradient structure, gradient system, evolutionary Gamma-convergence, energy-dissipation principle, variational evolution, relative entropy, large-deviation principle

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.

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