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

TL;DR

This paper compares two methods for deriving generalized Langevin equations from microscopic models, revealing fundamental differences in their adherence to fluctuation-dissipation relations in non-equilibrium systems.

Contribution

It demonstrates that projection operator and integration methods yield different GLEs in non-equilibrium contexts, highlighting the importance of the microscopic origin.

Findings

Mori-Zwanzig approach satisfies the generalized fluctuation-dissipation theorem

Integration method can violate the fluctuation-dissipation theorem

Numerical simulations confirm theoretical differences in non-equilibrium systems

Abstract

Generalized Langevin equations (GLEs) can be systematically derived via dimensional reduction from high-dimensional microscopic systems. For linear models the derivation can either be based on projection operator techniques such as the Mori-Zwanzig (MZ) formalism or by "integrating out" the bath degrees of freedom. Based on exact analytical results we show that both routes can lead to fundamentally different GLEs and that the origin of these differences is based inherently on the non-equilibrium nature of the microscopic stochastic model. The most important conceptional difference between the two routes is that the MZ result intrinsically fulfills the generalized second fluctuation-dissipation theorem while the integration result can lead to its violation. We supplement our theoretical findings with numerical and simulation results for two popular non-equilibrium systems: Time-delayed feedback control and the active Ornstein-Uhlenbeck process.