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- W1542303762 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 $delta tneqdelta t^{prime}$ 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 $TggDelta t$ longer than the considered lagtime $Delta t$." @default.
- W1542303762 created "2016-06-24" @default.
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- W1542303762 date "2012-07-10" @default.
- W1542303762 modified "2023-09-27" @default.
- W1542303762 title "Inhomogeneous ensembles of correlated random walkers" @default.
- W1542303762 doi "https://doi.org/10.48550/arxiv.1207.2242" @default.
- W1542303762 hasPublicationYear "2012" @default.
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