Repository: Freie Universit├Ąt Berlin, Math Department

On microscopic origins of generalized gradient structures

Liero, M. and Mielke, A. and Peletier, M. A. and Renger, M. (2017) On microscopic origins of generalized gradient structures. Discrete and Continuous Dynamical Systems - Series S, 10 (1).


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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.

Item Type:Article
Additional Information:SFB 1114 Preprint 07/2015 in WIAS Preprint No. 2148 Accepted for publication in STAMM DCDS-S proceedings
Subjects:Mathematical and Computer Sciences > Mathematics > Applied Mathematics
ID Code:1889
Deposited By: Ulrike Eickers
Deposited On:16 Mar 2016 18:34
Last Modified:12 Dec 2017 13:38

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