Random walk with resetting in a 1D chain

Abstract

If the classical model of random walks is added with the stochastic resetting to the starting point, then the whole process acquires new nontrivial features. In particular, there appears a non-equilibrium steady state. In addition, the mean first passage time (which is infinite in the absence of restarts) becomes finite and can be optimized by choosing a proper mean intermittence frequency r. It is shown that, in the case of random walks on the nodes of a one-dimensional chain, these effects essentially differ from their analogs within the classical continuous diffusion model. In particular, the asymptotes of the dependences of stationary node populations on r change from exponential to power ones. Similar qualitative and quantitative distinctions take place for the mean first passage time as well. In the case of a finite chain, the interesting effect of emergence and disappearance of a possibility of the minimization of this time, depending on the distance to a defined target, shows up.