Model and Statistical Analysis of the Motion of a Tired Random Walker in Continuum

TL;DR

This paper models a tired random walker with exponentially decreasing jump lengths, analyzing its motion in 1D, 2D, and 3D, revealing how tiring affects diffusion, return probability, and first passage times compared to normal walkers.

Contribution

It introduces a new model of a tired random walker with exponential decay in jump length and systematically studies its diffusive behavior across dimensions.

Findings

Diffusive behavior breaks down due to tiring in all dimensions.

Displacement distribution remains Gaussian in 1D, nonmonotonic in 2D and 3D.

Return probability decreases with tiring in 1D and 2D, but not in 3D.

Abstract

The model of a tired random walker, whose jump-length decays exponentially in time, is proposed and the motion of such a tired random walker is studied systematically in one, two and three dimensional contin- uum. In all cases, the diffusive nature of walker, breaks down due to tiring which is quite obvious. In one dimension, the distribution of the displace- ment of a tired walker remains Gaussian (as observed in normal walker) with reduced width. In two and three dimensions, the probability distribution of displacement becomes nonmonotonic and unimodal. The most probable displacement and the deviation reduces as the tiring factor increases. The probability of return of a tired walker, decreases as the tiring factor increases in one and two dimensions. However, in three dimensions, it is found that the probability of return almost insensitive to the tiring factor. The prob- ability distributions of first return time of a tired random walker do not show the scale invariance as observed for a normal walker in continuum. The exponents, of such power law distributions of first return time, in all three dimensions are estimated for normal walker. The exit probability and the probability distribution of first passage time are found in all three dimensions. A few results are compared with available analytical calculations for normal walker.