On the mean square displacement in Levy walks
Many physical and biological processes are modeled by "particles" undergoing L\'evy random walks. A feature of significant interest in these systems is the mean square displacement (MSD) of the particles. Long-time asymptotic approximations of the MSD have been established, via the Ta...
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Main Authors | , |
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Format | Journal Article |
Language | English |
Published |
25.03.2019
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Subjects | |
Online Access | Get full text |
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Summary: | Many physical and biological processes are modeled by "particles" undergoing
L\'evy random walks. A feature of significant interest in these systems is the
mean square displacement (MSD) of the particles. Long-time asymptotic
approximations of the MSD have been established, via the Tauberian Theorem, for
systems in which the distribution of the step durations is asymptotically a
power law of infinite variance. We extend these results, using elementary
analysis, and obtain closed-form expressions as well as power law bounds for
the MSD in equilibrium, and representations of the MSD as sums of super-linear,
linear, and sub-linear terms. We show that the super-linear components are
determined by the mean and asymptotics of the step durations, but that the
linear and sub-linear components (whose size has implications for the accuracy
of the asymptotic approximation) depend on the entire distribution function. |
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DOI: | 10.48550/arxiv.1903.10696 |