TY - BOOK

T1 - On microscopic origins of generalized gradient structures

AU - Liero, M.

AU - Mielke, A.

AU - Peletier, M.A.

AU - Renger, D.R.M.

PY - 2015

Y1 - 2015

N2 - 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 \Gamma-convergence, energy-dissipation principle, variational evolution, relative entropy, large-deviation principle

AB - 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 \Gamma-convergence, energy-dissipation principle, variational evolution, relative entropy, large-deviation principle

UR - http://arxiv.org/pdf/1507.06322.pdf

M3 - Report

T3 - arXiv

BT - On microscopic origins of generalized gradient structures

PB - s.n.

CY - s.l.

ER -