On microscopic origins of generalized gradient structures

Matthias Liero; Alexander Mielke; Mark A. Peletier; D. R. Michiel Renger

doi:10.3934/dcdss.2017001

Article Contents

2017, Volume 10, Issue 1: 1-35. Doi: 10.3934/dcdss.2017001

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On microscopic origins of generalized gradient structures

1.
Weierstra$\beta $-Institut für Angewandte Analysis und Stochastik, Mohrenstra$\beta $e 39, 10117 Berlin, Germany
2.
Department of Mathematics and Computer Science and Institute for Complex Molecular Systems (ICMS), Eindhoven University of Technology P.O. Box 513, 5600 MB Eindhoven, The Netherlands

* Corresponding author:Alexander Mielke
* Corresponding author:Alexander Mielke

Received: June 30, 2015

Revised: January 31, 2016

Published: February 2018

The research was partially supported by Einstein Stiftung Berlin, ERC AdG 267802, and DFG via SFB 1114

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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 $Γ$-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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Mathematics Subject Classification: Primary: 35K55, 35Q82, 49S05, 49J40, 49J45, 60F10, 60J25.

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Figure 1.1. For reversible, time-continuous Markov processes the large-deviation principle (LPD) of Section 2.4 provides a (generalized) gradient structure. This mapping commutes with taking the limit $\varepsilon\to 0$ and EDP-convergence, respectively.

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Figure 3.1. Left: Three-state Markov process with high rate of leaving state 2. Right: The limit for $\varepsilon \to 0$ gives a two-state Markov process.

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Figure 1.1. The potential $V$ along the reaction path $\Upsilon =[0,7]$.

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