Novel anomalous diffusion phenomena of underdamped Langevin equation with random parameters

TL;DR

This paper investigates how random parameters in an underdamped Langevin system affect anomalous diffusion and ergodicity, revealing that randomness in relaxation time and initial velocity can significantly enhance diffusion behavior.

Contribution

It introduces a novel analysis of the impact of random relaxation timescale and initial velocity on diffusion and ergodic properties in Langevin systems with complex environments.

Findings

Heavy-tailed distribution of relaxation time suppresses velocity correlation decay.

Random initial velocity can enhance diffusion depending on its distribution.

Random parameters significantly influence system dynamics beyond finite moments.

Abstract

The diffusion behavior of particles moving in complex heterogeneous environment is a very topical issue. We characterize particle's trajectory via an underdamped Langevin system driven by a Gaussian white noise with a time dependent diffusivity of velocity, together with a random relaxation timescale $\tau$ to parameterize the effect of complex medium. We mainly concern how the random parameter $\tau$ influences the diffusion behavior and ergodic property of this Langevin system. Besides, the comparison between the fixed and random initial velocity $v_0$ is conducted to show the effect of different initial ensembles. The heavy-tailed distribution of $\tau$ with finite mean is found to suppress the decay rate of the velocity correlation function and promote the diffusion behavior, playing a competition role to the time dependent diffusivity. More interestingly, a random $v_0$ with a specific distribution depending on random $\tau$ also enhances the diffusion. Both the random parameters $\tau$ and $v_0$ influence the dynamics of the Langevin system in an non-obvious way, which cannot be ignored even they has finite moments.