Transport Regimes of Underdamped Brownian Particles in a Tilted Washboard Potential

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" andrunning" 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].