Dynamic Coarse-Graining via Large-Deviation Theory

Abstract

Coarse-graining is the art of a useful and consistent reduction of information and, in physics, it addresses the question: Given two levels of description of a system, how can we find a model at the more macroscopic (less detailed) level starting from a model at the more microscopic (more detailed) level? In this work we formulate this question as the problem of determination of the dissipative structure of the more macroscopic model. Since the works by Onsager it has become clear that the dissipative component of many evolution equations of nonequilibrium thermodynamics has the structure of a gradient flow, the chief example being hydrodynamics: the dynamics is given by the product of a force (the derivative of a driving function) and a phenomenological matrix. The phenomenological matrix can be computed by studying the fluctuations of the macroscopic model in the form of a diffusion process, and in particular it is equal to the the diffusion tensor. This is the essence of the fluctuation-dissipation theorem of the second kind (FDT) and gives rise to the method of Green-Kubo relations. However, this scheme does not exhaust the class of nonequilibrium dissipative systems: for example, chemical reactions are much better described by jump processes rather than by diffusion processes. We thus need to extend the FDT. For this reason, the introductory chapters of the thesis are devoted to a possible definition of the terms "fluctuation" and "dissipation". Fluctuations are formalised by the theory of large deviations, which is the mathematical language of statistical mechanics: thermodynamic potentials are directly related to large-deviation rate functions. Dissipation is formulated in terms of dissipation potentials, which – together with a driving function – define generalised gradient flows, a nonlinear extension of gradient flows. Following the work [A. Mielke, M. A. Peletier and D. R. M. Renger. 'On the Relation between Gradient Flows and the Large-Deviation Principle, with Applications to Markov Chains and Diffusion'. In: Potential Analysis 41.4 (2014), pp. 1293-1327], we extend the FDT from diffusion processes to a wider class of stochastic processes, namely the class of Markov processes that satisfy detailed balance and a large-deviation principle. The generalized FDT characterises the deterministic limit of a sequence of Markov processes as a generalized gradient flow: the dissipation potential is derived from the large-deviation dynamic rate function of the Markov processes, and the driving function is the rate function of their stationary distributions. The most important consequence, which constitutes the main contribution of our work, is the formulation of a broader theory of coarse-graining that is valid not just for hydrodynamic-like, but also for rare-event systems, which are described by jump processes rather than by diffusion processes. We test the new method, both theoretically and numerically, in the context of the passage from the diffusion in a double-well potential to the jump process that models the simple reaction A<->B, the Kramers escape problem.