Slow Kinetics of Brownian Maxima

Abstract

We study extreme-value statistics of Brownian trajectories in one dimension. We define the maximum as the largest position to date and compare maxima of two particles undergoing independent Brownian motion. We focus on the probability P (t) that the two maxima remain ordered up to time t, and find the algebraic decay P ∼ t −β with exponent β = 1/4. When the two particles have diffusion constants D1 and D2, the exponent depends on the mobilities, β = 1 π arctan D2/D1. We also use numerical simulations to investigate maxima of multiple particles in one dimension and the largest extension of particles in higher dimensions.PACS numbers: 05.40. Jc, 05.40.Fb, 02.50.Cw, 02.50.Ey Consider a pair of particles undergoing independent Brownian motion in one dimension [1]. These two particles do not meet with probability that decays as t −1/2 in the long-time limit. This classical first-passage behavior holds for Brownian particles with arbitrary diffusion constants. It holds even for particles undergoing symmetric Lévy flights [2,3], and has numerous applications [3,4]. Here, we generalize this ubiquitous first-passage behavior to maxima of Brownian particles. Figure 1 shows that the maximal position of each particle forms a staircase and it illustrates that unlike the position, the maximum is a non-Markovian random variable [5,6]. We find that two such staircases do not intersect with probability P that is inversely proportional to the one-fourth power of time, P ∼ t −1/4 , in the long-time limit. If the particles move with diffusion constants D 1 and D 2 , the two maxima remain ordered during the time interval (0, t) with the slowly-decaying probabilityIn this letter, we obtain this result analytically and investigate numerically related problems involving multiple maxima and diffusion in higher dimensions. Anomalous relaxation with nontrivial persistence exponents [7][8][9], enhanced transport due to disorder [10,11], and anomalous diffusion due to exclusion [12,13] are dynamical phenomena that were recently demonstrated in experiments involving Brownian particles. Understanding the nonequilibrium statistical physics of these diffusion processes is closely intertwined with the characteristic behavior of extreme fluctuations and the statistics of extreme values [14][15][16][17][18][19].We first establish Eq. (1) for two Brownian particles having the same diffusion constant D. Let us denote the positions of the particles at time t by x 1 (t) and x 2 (t), and without loss of generality, we assume x 1 (0) > x 2 (0). We define the maximum of the first particle, m 1 (t), to be its rightmost position up to time t; similarly, m 2 (t) is the maximal position of the second particle. Our goal is to find the probability P (t) that the two maxima remain ordered m 1 (τ ) > m 2 (τ ) for all 0 ≤ τ ≤ t.The two maxima remain ordered if and only if m 1 (τ ) > x 2 (τ ) at all times 0 ≤ τ ≤ t. Hence, to find P , there is no need to keep track of the maximum m 2 , and it suffices to consider only the position x 2 . As a further simplificat...

Record statistics of a strongly correlated time series: random walks and Lévy flights

“…as demonstrated by numerical simulations [130]. The only analytical result concerns two random walks for which…”

“…We recall that the average number of records is given by the sum in (130). This sum over k is dominated by the values of k ∼ O(N ) which are thus large, when N 1 [86].…”

Maxima of two random walks: universal statistics of lead changes

“…This distribution would imply that the probability of having no lead change decays as exp[−(ln t)/π] ∼ t −1/π , that is, slower than the f 0 ∼ t −1/4 behavior which has been established analytically [13,14]. It is straightforward to generalize equation (4) to the situation where the two random walks have different diffusion coefficients, denoted by D 1 and D 2 .…”

“…We now focus on the probability f n (t) to have exactly n lead changes until time t. For two identical random walks, the probability that there are no lead changes decays as f 0 ∼ t −1/4 in the long-time limit [13,14]. Since f n (t) is the probability that the (n + 1)-st lead change takes place after time t, the probability density for the (n + 1)-st lead change to occur at time t is given by −df n /dt.…”

“…1). The probability that one maximum exceeds another during the time interval (0, t) was investigated only recently [13,14]. Here we explore additional features of the interplay between two Brownian maxima.…”