A - Data analysis
from Appendices
Published online by Cambridge University Press: 05 August 2012
Summary
A criterion for inferring superdiffusion
Here we review the criterion for establishing superdiffusion reported in [396]. We argue that this is a necessary but not a sufficient condition.
Consider two-dimensional correlated random walk (CRW) models, in which persistence (or directional memory) is controlled by the probability distribution of the relative turning angles. We define the turning angle θj as the difference between the angles of successive step vectors rj+1 and rj. If the turning angles are uniformly and independently distributed, then the path follows the usual uncorrelated Brownian random walk. However, if the turning angles are nonuniformly distributed, then even for independently and identically distributed turning angles, the resulting random walk step vectors rj may be autocorrelated. In the extreme case, if the probability density function (pdf) for the turning angles is given by a Dirac δ-function at θ = 0, the resulting walk is a straight line; i.e., the motion is ballistic.
However, generally, CRW models cannot have scale invariance. Instead, CRWs possess a characteristic scale (or time τ) associated with the exponentially decaying correlations always present in Markov processes. To obtain τ, we define an adimensional two-point correlation function in terms of the random walk step vectors rj as C(|j - i|) ≡ 〈rj · ri〉/〈rjri〉, where i, j are integer indices representing time, assuming constant speed. The notation 〈·〉 denotes averaging (either averaging along the walk or ensemble averaging, depending on the context).
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- The Physics of ForagingAn Introduction to Random Searches and Biological Encounters, pp. 131 - 135Publisher: Cambridge University PressPrint publication year: 2011