Statistical Mechanics of Coarse-Graining

Abstract

We review the concepts of the theory of coarse-graining and its mathematical background based on the projection operator technique. The objective of the theory is to derive the Fokker–Planck equation that governs the probability distribution of the coarse-grained variables. The essential practical problem of an explicit coarse-graining procedure from the microscopic dynamics, which is the high dimensionality of state space, is pinpointed. This problem enforces one to model the objects appearing in the Fokker–Planck equation. In this case, the program of coarse-graining helps in producing strong conditions on the possible forms of these objects. In particular, we review the stringent GENERIC structure that emerges when the dynamical invariants can be expressed as functions of the coarse-grained variables. Finally, we illustrate how the GENERIC framework may help in the task of inventing new discrete models for the simulation of complex fluids that are thermodynamically consistent by construction.