Persistent Random Walks and Reactions

2 Reactions and Transport Diffusion, Inertia, tind Subdiffusion 

In the correlated or persistent random walk [474], a particle or individual takes steps of length Ax and duration At. The particle continues in its previous direction with probability a = — fxAt and reverses direction with probability = fiAt. In the continuum limit Ax 0 and At 0, such that 

The particles travel with speed y and turn with frequency jx. The persistent random walk is characterized by two parameters, in contrast to the ordinary random walk or Brownian motion, which is completely characterized by the diffusion coefficient D. The persistent random walk spans the whole range of dispersal, from ballistic motion, in the limit /r 0, to diffusive motion, in the limit y oo, p. oo, such that lim y 2p = Z) = constant. The total density of the dispersing particles is given by 

Differentiating (2.29) with respect to t and (2.30) with respect to x and eliminating the mixed second derivatives, we obtain the telegraph equation 

Besides these practical considerations, describing the motion of particles or individuals by a persistent random walk has several advantages from a theoretical viewpoint (i) The persistent random walk is a generalization of Brownian motion it contains the latter as a limiting case, see above, (ii) The persistent random walk overcomes the pathological feature of Brownian motion or the diffusion equation discussed above it fulfills the physical requirement of bounded velocity, (iii) The persistent random walk provides a unified treatment that covers the whole range of transport, from the diffusive limit to the ballistic limit. 

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