Viscoelastic active diffusion governed by nonequilibrium fractional Langevin equations: underdamped dynamics and ergodicity breaking

TL;DR

This paper explores the complex nonequilibrium dynamics and ergodicity breaking in a viscoelastic active system modeled by a fractional Langevin equation with active noise, revealing ultraweak ergodicity breaking and long-lived nonergodic states.

Contribution

It provides analytical solutions and detailed analysis of active fractional Langevin equations, highlighting new ergodicity-breaking phenomena and the interplay of active noise with viscoelastic memory effects.

Findings

Active noise causes ultraweak ergodicity breaking.

Long-lived nonergodic states occur as H approaches 1.

System eventually becomes ergodic at infinite time.

Abstract

In this work, we investigate the active dynamics and ergodicity breaking of a nonequilibrium fractional Langevin equation (FLE) with a power-law memory kernel of the form $K(t)\sim t^{-(2-2H)}$, where $1/2<H<1$ represents the Hurst exponent. The system is subjected to two distinct noises: a thermal noise satisfying the fluctuation-dissipation theorem and an active noise characterized by an active Ornstein-Uhlenbeck process with a propulsion memory time $\tau_\mathrm{A}$. We provide analytic solutions for the underdamped active fractional Langevin equation, performing both analytical and computational investigations of dynamic observables such as velocity autocorrelation, the two-time position correlation, ensemble- and time-averaged mean-squared displacements (MSDs), and ergodicity-breaking parameters. Our results reveal that the interplay between the active noise and long-time viscoelastic memory effect leads to unusual and complex nonequilibrium dynamics in the active FLE systems. Furthermore, the active FLE displays a new type of discrepancy between ensemble- and time-averaged observables. The active component of the system exhibits ultraweak ergodicity breaking where both ensemble- and time-averaged MSDs have the same functional form with unequal amplitudes. However, the combined dynamics of the active and thermal components of the active FLE system are eventually ergodic in the infinite-time limit. Intriguingly, the system has a long-standing ergodicity-breaking state before recovering the ergodicity. This apparent ergodicity-breaking state becomes exceptionally long-lived as $H\to1$, making it difficult to observe ergodicity within practical measurement times. Our findings provide insight into related problems, such as the transport dynamics for self-propelled particles in crowded or polymeric media.