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.

References in corpus (11)

• Non-Equilibrium Statistical Mechanics of Anharmonic Chains Coupled to Two Heat Baths at Different Temperatures
• Non-Equilibrium Statistical Mechanics of Strongly Anharmonic Chains of Oscillators
• Exponential Convergence to Non-Equilibrium Stationary States in Classical Statistical Mechanics
• Data-driven model reduction, Wiener projections, and the Koopman-Mori-Zwanzig formalism
• A Dynamic Subgrid Scale Model for Large Eddy Simulations Based on the Mori-Zwanzig Formalism
• The Derivation and Approximation of Coarse-grained Dynamics from Langevin Dynamics
• Hypocoercivity for Kolmogorov backward evolution equations and applications
• Computation of the Memory Functions in the Generalized Langevin Models for Collective Dynamics of Macromolecules
• Hilbert space hypocoercivity for the Langevin dynamics revisited
• Hypoellipticity and the Mori-Zwanzig formulation of stochastic differential equations
• Effective Mori-Zwanzig equation for the reduced-order modeling of stochastic systems