Nonergodic solutions of the generalized Langevin equation

TL;DR

This paper investigates conditions under which the generalized Langevin equation admits non-ergodic solutions, revealing that non-ergodicity can occur even with finite memory and normal diffusion.

Contribution

It demonstrates the existence of non-ergodic solutions in the generalized Langevin equation beyond the superlinear diffusion regime, including cases with finite memory and normal diffusion.

Findings

Non-ergodic solutions can occur with zero integral friction.

Non-ergodic solutions exist even when the memory function's integral is finite.

Non-ergodicity is possible outside the superlinear diffusion regime.

Abstract

It is known that in the regime of superlinear diffusion, characterized by zero integral friction (vanishing integral of the memory function), the generalized Langevin equation may have non-ergodic solutions which do not relax to equilibrium values. It is shown that the equation may have non-ergodic (non-stationary) solutions even if the integral of the memory function is finite and diffusion is normal.