Transport Regimes of Underdamped Brownian Particles in a Tilted Washboard Potential

TL;DR

This study explores how underdamped Brownian particles in a tilted washboard potential transition between different diffusion regimes, revealing temperature thresholds, inertial effects, and oscillatory behaviors that influence mobility and diffusivity.

Contribution

It provides a detailed phase diagram and identifies inertial effects as key factors in anomalous diffusion and negative differential mobility in such systems.

Findings

Identified a temperature threshold for diffusive regime transition.

Observed negative differential mobility at low temperatures and biases.

Linked inertial effects to oscillatory velocity spectra and diffusion behaviors.

Abstract

In this paper, a comprehensive examination of the temperature- and bias-dependent diffusion regimes of underdamped Brownian particles is presented. A temperature threshold for a transition between anomalous and normal diffusive behaviors is located, yielding a new phase diagram for the system. In the low-temperature regime, the system exhibits an apparent negative differential mobility due to persistent, long-time subdiffusion at low-bias; at high temperature (or critical bias,) the system rapidly approaches normal diffusion below an intermediate barrier height, $U_o \sim k_B T$. By consideration of numerical results, comparison to the overdamped case, and the related Kramers multistable escape problems, it is demonstrated that the low-bias non-monotonic temperature dependence of the diffusivity, persistent subdiffusion, and negative differential mobility can be traced to inertial effects, which are evident in the oscillatory modes of the velocity power spectra at low bias. In the giant diffusion regime, the velocity power spectra exhibit coupling between the ``locked" and ``running" states, with a characteristic frequency corresponding to the principal frequency of the limit cycles of a damped, driven plane pendulum near critical bias. Non-linear second harmonic generation, corresponding to oscillatory transient anomalous diffusivity, is observed with increasing bias and decreasing temperature, further emphasizing that the low-noise diffusion problem converges to noise-free dynamics, complementing analytic results for the average velocity [L. Cheng and N.K. Yip, Physica D, 2015].