Memory-induced slow relaxation in the generalized Langevin equation

Abstract

We investigate the relaxation dynamics of the generalized Langevin equation (GLE) with an exponential memory kernel. Our analysis reveals that the relaxation time to equilibrium scales inversely with the square of the memory decay rate λ, i.e., τrelax ∝ λ−2. To understand the origin of this scaling, we analyze the escape dynamics from a potential well and find that the mean escape time follows the same λ−2 dependence. This indicates that the slow relaxation arises from memory-suppressed fluctuations that delay escape events. To explain these findings, we reformulate the GLE using an auxiliary variable, embedding the non-Markovian dynamics in a higher-dimensional Markovian system. This approach provides a clear physical picture of how memory influences noise, diffusion, and relaxation in non-Markovian systems.