Average time spent by Lévy flights and walks on an interval with absorbing boundaries

Abstract

We consider a Lévy flyer of order alpha that starts from a point x(0) on an interval [O,L] with absorbing boundaries. We find a closed-form expression for the average number of flights the flyer takes and the total length of the flights it travels before it is absorbed. These two quantities are equivalent to the mean first passage times for Lévy flights and Lévy walks, respectively. Using fractional differential equations with a Riesz kernel, we find exact analytical expressions for both quantities in the continuous limit. We show that numerical solutions for the discrete Lévy processes converge to the continuous approximations in all cases except the case of alpha-->2, and the cases of x(0)-->0 and x(0)-->L. For alpha>2, when the second moment of the flight length distribution exists, our result is replaced by known results of classical diffusion. We show that if x(0) is placed in the vicinity of absorbing boundaries, the average total length has a minimum at alpha=1, corresponding to the Cauchy distribution. We discuss the relevance of this result to the problem of foraging, which has received recent attention in the statistical physics literature.

Walk dimension for light in complex disordered media

“…which is a generalized Ohm's law valid in the limit of very large L, where finite-size effects can be neglected [17,43].…”

Optimal Intermittence in Search Strategies under Speed-Selective Target Detection

“…Formally, this can be studied as a first-passage problem in a disordered media. For such situations it has been shown both analytically and numerically [1][2][3][4] that Lévy flights represent an optimal search strategy since they minimize the mean time for detection or mean first-passage time. The alternation between small jumps in a given region (clustering) and relatively frequent long jumps typical of Lévy motion seems to represent the key to this optimality, even in those cases when the Lévy process is superposed to other types of motion [5].…”

Universal coefficient of the exact correlator of a large-Nmatrix field theory

“…The result is 1) which is the mean first-passage time of a Lévy flight in one dimension, calculated via the corresponding continuum theory, namely the anomalous Laplace equation. Since the discussion in Reference [8] may not be easily accessible to readers working in quantum field theory, we present a short derivation of (4.1) below. We write the square root of the one-dimensional Laplacian (3.2) as (with b = −1, c = 1)…”

“…An expression which is proportional to the right-hand-side of (3.6) was evaluated in Reference [8]. The result is 1) which is the mean first-passage time of a Lévy flight in one dimension, calculated via the corresponding continuum theory, namely the anomalous Laplace equation.…”

“…We will show that C 2 = 1/16π. In fact, this quantity has already been evaluated in context of Lévy flights [8]; it is proportional to the mean-first passage time in one dimension.…”