Inhomogeneous ensembles of correlated random walkers

TL;DR

This paper explores inhomogeneous ensembles of correlated random walkers, revealing how their statistical properties, such as displacement distributions and mean squared displacement, depend on the distribution of persistence probabilities and step lengths, with analytical and scaling insights.

Contribution

It introduces a model of inhomogeneous correlated random walks with variable persistence and step lengths, deriving analytical results and showing how different ensembles can share statistical properties through scaling transformations.

Findings

Displacement probability density can be leptocurtic.

Mean squared displacement can follow a fractional power law.

Different ensembles can have equivalent long-term statistical properties.

Abstract

Discrete time random walks, in which a step of random sign but constant length $\delta x$ is performed after each time interval $\delta t$, are widely used models for stochastic processes. In the case of a correlated random walk, the next step has the same sign as the previous one with a probability $q \neq 1/2$. We extend this model to an inhomogeneous ensemble of random walkers with a given distribution of persistence probabilites $p(q)$ and show that remarkable statistical properties can result from this inhomogenity: Depending on the distribution $p(q)$, we find that the probability density $p(\Delta x, \Delta t)$ for a displacement $\Delta x$ after lagtime $\Delta t$ can have a leptocurtic shape and that mean squared displacements can increase approximately like a fractional powerlaw with $\Delta t$. For the special case of persistence parameters distributed equally in the full range $q \in [0,1]$, the mean squared displacement is derived analytically. The model is further extended by allowing different step lengths $\delta x_j$ for each member $j$ of the ensemble. We show that two ensembles $[\delta t, {(q_j,\delta x_j)}]$ and $[\delta t^{\prime}, {(q^{\prime}_j,\delta x^{\prime}_j)}]$ defined at different time intervals $\delta t\neq\delta t^{\prime}$ can have the same statistical properties at long lagtimes $\Delta t$, if their parameters are related by a certain scaling transformation. Finally, we argue that similar statistical properties are expected for homogeneous ensembles, in which the parameters $(q_j(t),\delta x_j(t))$ of each individual walker fluctuate temporarily, provided the parameters can be considered constant for time periods $T\gg\Delta t$ longer than the considered lagtime $\Delta t$.