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.
Highlights
We consider evolution equations u = V (t, u) that are generated by gradient systems (GS)
In all our three examples we find the surprising result that the energy-dissipation principle (EDP)-limit is exactly the entropic GS of the limiting Markov process
We consider a compact metric space S and denote by Prob(S) the subset of probability Radon measures on S equipped with the narrow convergence ⇀∗ defined by duality with continuous, bounded functions
Summary
The three main models in this work (i.e. the ODE, the membrane, and the reaction-to-diffusion model in Sections 3.3.2, 4, and 5, respectively) can be seen as Kolmogorov forward equations for naturally associated Markov processes. In all our three examples we find the surprising result that the EDP-limit is exactly the entropic GS of the limiting Markov process This means that applying the described large-deviation principle and taking the limit ε → 0 (either on the level of Markov semigroups or as EDP-convergence for GS) commute, see Figure 1.1. This result appears naturally, if we use representation (1.6) of the rate function I giving. We do not give the full analytical details in terms of estimates and convergences in the proper functional spaces, but rather highlight the structures and manipulations needed to understand the corresponding limit procedures
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