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arXiv · 1207.2242

Inhomogeneous ensembles of correlated random walkers

F. Stadler·C. Metzner·J. Steinwachs·B. Fabry
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Abstract

Discrete time random walks, in which a step of random sign but constant length $δx$ is performed after each time interval $δt$, 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 $p(Δx, Δt)$ for a displacement $Δx$ after lagtime $Δt$ can have a leptocurtic shape and that mean squared displacements can increase approximately like a fractional powerlaw with $Δt$. 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 $δx_j$ for each member $j$ of the ensemble. We show that two ensembles $[δt, {(q_j,δx_j)}]$ and $[δt^{\prime}, {(q^{\prime}_j,δx^{\prime}_j)}]$ defined at different time intervals $δt\neqδt^{\prime}$ can have the same statistical properties at long lagtimes $Δt$, 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 $(q_j(t),δx_j(t))$ of each individual walker fluctuate temporarily, provided the parameters can be considered constant for time periods $T\ggΔt$ longer than the considered lagtime $Δt$.

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BibTeXRIS

F. Stadler, C. Metzner, J. Steinwachs, B. Fabry. 2012-07-10. Inhomogeneous ensembles of correlated random walkers. https://arxiv.org/abs/1207.2242

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