Arcsine Law and Multistable Brownian Dynamics in a Tilted Periodic Potential

Abstract

Multistability is one of the most important phenomena in dynamical systems, e.g. bistability enables the implementation of logic gates and therefore computation. Recently multistability has attracted a greatly renewed interest related to memristors and graphene structures. We investigate tristability in velocity dynamics of a Brownian particle subjected to a tilted periodic potential. It is demonstrated that the origin of this effect is attributed to extreme value statistics in the form of arcsine law for the velocity dynamics at the zero temperature limit. We analyze the impact of thermal fluctuations and construct the phase diagram for the stability of the velocity dynamics. It suggests an efficient strategy to control the multistability by changing solely the force acting on the particle or temperature of the system. Our findings for the paradigmatic model of nonequilibrium statistical physics apply to, inter alia, Brownian motors, Josephson junctions, cold atoms dwelling in optical lattices and colloidal systems.

Conundrum of weak noise limit for diffusion in a tilted periodic potential

“…Finally, in Ref. [52] it is shown that the real-time velocity dynamics in this system is not bistable but rather multistable.…”

“…Consequently, x(t) ∼ v t, the leading term in the mean square displacement is ∆x 2 (t) ∼ t 2 and one observes the persistent ballistic diffusion with D(t) → ∞. Its constancy is guaranteed via the strong ergodicity breaking [52,55], i.e. there are two mutually inaccessible attractors for the average velocity of the particle in the phase space of the system.…”