Persistence and First-Passage Properties in Non-equilibrium Systems

Abstract

In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.

Statistics of superior records

“…FIG. 2: (Color online) The scaling behavior (11). Shown are the normalized polynomials ΦN = FN /SN versus the variable s for N ≤ 4.…”

“…Essentially, this is as an evolution equation with the sequence length N playing the role of time. By substituting the scaling form (11) into the evolution equation (12) and by using the algebraic decay (1), we find that the scaling function Φ(s) obeys the differential equation…”

“…Studies of extreme value statistics typically focus on average and extremal properties of the distribution of extreme values [8,9]. Yet so far, first passage and persistence properties (see [10,11] and references therein) have not received significant attention in the context of extreme values.…”

Sampling fractional Brownian motion in presence of absorption: A Markov chain method

“…with θ being the persistence exponent known to be θ = 1 − H [3,24,25]. Hence, for the case L = 10 7 and H = 1/4, which we study here, we obtain P 0 (L) ≈ 10 −5 .…”

First Passage in Conical Geometry and Ordering of Brownian Particles

“…This process ends when the wealth of the gambler reaches either zero or a high threshold. 6,7 First-passage properties of multiple random walks 8,9 and particles undergoing Brownian motion [10][11][12] in one dimension are the subject of ongoing research and underlie dynamics of spin chains, [13][14][15] voting processes, [16][17][18] urn model, 19 and unraveling of knots. 20 Many of these firstpassage processes are closely related to the ordering of a set of independent Brownian particles on a line.…”