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Stochastic processes
Abstract:
What happens when the time evolution of a fluctuating interface is interrupted by resetting to a given initial configuration after random time intervals tau distributed as a power-law similar to tau(-(1+alpha)); alpha > 0? For an interface of length L in one dimension, and an initial flat configuration, we show that depending on alpha, the dynamics in the limit L -> infinity exhibit a spectrum of rich long-time behavior. It is known that without resetting, the interface width grows unbounded with time as t(beta) in this limit, where beta is the so-called growth exponent characteristic of the universality class for a given interface dynamics. We show that introducing resetting leads to fluctuations that are bounded at large times for alpha > 1. Corresponding to such a reset-induced stationary state is a distribution of fluctuations that is strongly non-Gaussian, with tails decaying as a power-law. The distribution exhibits a distinctive cuspy behavior for a small argument, implying that the stationary state is out of equilibrium. For alpha < 1, on the contrary, resetting to the flat configuration is unable to counter the otherwise unbounded growth of fluctuations in time, so that the distribution of fluctuations remains time dependent with an ever-increasing width, even at long times. Although stationary for alpha > 1, the width of the interface grows forever with time as a power-law for 1 < alpha < alpha((w)), and converges to a finite constant only for larger a, thereby exhibiting a crossover at alpha((w)) = 1 + 2 beta. The time-dependent distribution of fluctuations exhibits for alpha < 1 and for small arguments further interesting crossover behavior from cusp to divergence across alpha((d)) = 1 - beta. We demonstrate these results by exact analytical results for the paradigmatic Edwards-Wilkinson (EW) dynamical evolution of the interface, and further corroborate our findings by extensive numerical simulations of interface models in the EW and Kardar-Parisi-Zhang universality class.
