Motion of a random walker in a quenched power law correlated velocity field

TL;DR

This paper analyzes the motion of a random walker in a quenched power law correlated velocity field, revealing fractional Brownian motion behavior and calculating persistence probabilities with analytical and simulation methods.

Contribution

It introduces a modified MdM model with long-range correlations and analytically derives the Hurst exponent and persistence properties of the walker's motion.

Findings

Motion is fractional Brownian with H = max[1/2, (1- alpha/4), (1-d/4)]

Derived disorder-averaged persistence probability over time

Identified marginal behavior lines in parameter space d and alpha

Abstract

We study the motion of a random walker in one longitudinal and d transverse dimensions with a quenched power law correlated velocity field in the longitudinal x-direction. The model is a modification of the Matheron-de Marsily (MdM) model, with long-range velocity correlation. For a velocity correlation function, dependent on transverse co-ordinates y as 1/(a+|{y_1 - y_2}|)^alpha, we analytically calculate the two-time correlation function of the x-coordinate. We find that the motion of the x-coordinate is a fractional Brownian motion (fBm), with a Hurst exponent H = max [1/2, (1- alpha/4), (1-d/4)]. From this and known properties of fBM, we calculate the disorder averaged persistence probability of x(t) up to time t. We also find the lines in the parameter space of d and alpha along which there is marginal behaviour. We present results of simulations which support our analytical calculation.