Einstein-Helfand form for transport coefficients from coarse-grained descriptions

Abstract

We revisit the statistical mechanics problem of coarse-graining a system that at a detailed level is described by an already coarse-grained dynamics. The dynamics at the more detailed level is described by a Fokker-Planck equation instead of the Liouville equation. The method generalizes Zwanzig theory of projection operators and produces a friction matrix in terms of a correlation function that is not manifestly an autocorrelation. Therefore, from this expression, it is not obvious that the friction matrix is definite positive. We show that the Green-Kubo transport matrix can be written in the Einstein-Helfand form, which is manifestly positive definite. We also discuss the role of time reversal and detailed balance in the coarse-grained dynamics.