Novel anomalous diffusion phenomena of underdamped Langevin equation with random parameters

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 τ to parameterize the effect of complex medium. We mainly concern how the random parameter τ 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 τ 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 τ also enhances the diffusion. Both the random parameters τ and v 0 influence the dynamics of the Langevin system in an non-obvious way, which cannot be ignored even they has finite moments.