A simple hypocoercivity analysis for the effective Mori-Zwanzig equation

Abstract

We provide a simple hypocoercivity analysis for the effective Mori-Zwanzig equation governing the time evolution of noise-averaged observables in a stochastic dynamical system. Under the hypocoercivity framework mainly developed by Dolbeault, Mouhot and Schmeiser and further extended by Grothaus and Stilgenbauer, we prove that under the same conditions which lead to the geometric ergodicity of the Markov semigroup e tK , the Mori-Zwanzig orthogonal semigroup e tQKQ is also geometrically ergodic, provided that P = I − Q is a finite-rank, orthogonal projection operator in a certain Hilbert space. The result is applied to the widely used Mori-type effective Mori-Zwanzig equations in the coarse-grained modeling of molecular dynamics and leads to exponentially decaying estimates for the memory kernel and the fluctuation force.

Non-Markovian systems out of equilibrium: exact results for two routes of coarse graining

“…Another systematic route to reduce the dimensionality of a system is given by the projection operator formalism [8][9][10]43]. In the following, we will focus on the MZ formalism, which was originally developed for Hamiltonian dynamics, but has been shown to also be applicable to stochastic systems [22][23][24][44][45][46]. In the next subsection, we will shortly recap the important properties of the MZ formalism along the lines of reference [24], and then explicitly evaluate the formal expressions for the specific SDE discussed in this work.…”