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

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.

Nonequilibrium relaxation of a trapped particle in a near-critical Gaussian field

“…we obtain an effective equation for the dynamics of the particle alone. Note that projecting the fast degrees of freedom over the dynamics of the tracer particle adopting the Mori-Zwanzig scheme [23,24], which renders a linear equation, may lead to uncontrolled results in the present case, because the effective particle dynamics is actually nonlinear [25,26]. We follow instead Ref.…”