Fractional positional jumps in stochastic systems with tilted periodic double-well potentials

Abstract

We present a theoretical investigation of the stochastic dynamics of a damped particle in a tilted periodic potential with a double well per period. By applying the matrix continued fraction technique to the Fokker-Planck equation in conjunction with the full counting statistics and master equation approaches, we determine the rates of specific processes contributing to the system's overall dynamics. At low temperatures, the system can exhibit one running state and two distinct locked metastable states. We focus primarily on two aspects: the dynamics of positional jumps, which are rare thermally induced particle jumps over potential maxima, and their impact on the overall velocity noise; and the retrapping process, involving the transition from the running to the locked metastable states. We demonstrate the existence of fractional (in units of $2π$) positional slips that differ qualitatively from conventional $2π$ jumps observed in single-well systems. Fractional positional slips significantly influence the system dynamics even in regimes dominated by dichotomous-like switching between running and locked states. Furthermore, we introduce a simple master equation approach that proves effective in analyzing various stages of the retrapping process. Interestingly, our analysis shows that even for a system featuring a well-developed double-well periodic potential, there exists a broad parameter range where the stochastic dynamics can be accurately described by an effective single-well periodic model. The techniques introduced here allow for valuable insights into the complex behavior of the system, offering avenues for understanding and controlling its steady-state and transient dynamics, which go beyond or can be complementary to direct stochastic simulations.