Random walk persistent

RANDOM WALK DIFFUSION PERSISTENCE TIME TWIDDLING... 

Exercise. Should the random walk with persistence (1.7.8) be called a Markov process [The answer is given in IV.5.]... 

The random walk model may be generalized by introducing a statistical correlation between two successive steps, in such a way that the probability a for a step in the same direction as the previous step differs from the probability ft for a step back ( random walk with persistence ). In this case... 

Exercise. The transition matrix (5.5) for the random walk with persistence acts in too large a space, because only the states with m = n + 1 have a meaning. Find a simpler reduction of the random walk with persistence to a Markov chain by adding a second variable Y2 which takes only two values. 

Exercise. The mean square distance of the random walk with persistence can easily be found by the following alternative method, compare (1.7.8). It is equal to <(X + X2 + + Xr)2 >, where each Xk takes the values +1, and (XkXk+1) 0. 

Exercise. Show for the one-dimensional random walk with persistence that the distribution approaches a Gaussian. 

By correctly scaling po, a diffusion equation for the persistent random walk can be obtained from the recurrence relations of Eq. (551). One of the possible variants of scaling is examined when changing to continuous variables x and t. 

Cells migrating in a chemoattractant gradient exhibit a biased random walk, with migration persisting in the direction of the gradient because they turn less and less often than during random migration. If a cell is... 

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... 

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. 

If the particles moving according to a persistent random walk react with each other, the evolution equations for the densities, (2.25) and (2.26), must be modified... 

For n-variable systems, the evolution equations for persistent random walks with reaction read, i = 1,... 

Front Propagation in Persistent Random Walks with Reactions 169... 

As discussed in Sect. 2.2, persistent random walks provide a mesoscopic description of reaction-transport systems with inertia. This approach provides another opportunity to explore the effects of a finite velocity in the transport mechanism on propagating fronts. We consider two cases. The first corresponds to reaction walks where the kinetic terms do not depend on the direction of the particles. This corresponds to choosing /f = 1/2 in (2.38), and persistent random walks with such kinetics are called direction-independent reaction walks (DIRW). The second case corresponds to walks where reactions occur only between particles with opposite velocities. We call such systems direction-dependent reaction walks (DDRWs). 

If we replace Brownian motion by its simplest generalization, the persistent random walk, we obtain direction-independent reaction walks as the simplest generalization of reaction-diffusion equations. Both describe chemical reactions in the reaction-limited or activation-controlled regime. However, the activation barrier is only implicitly taken into account it is incorporated into the kinetic coefficients... 

As a third approach, we use persistent random walks with reactions, see Sect. 2.2.3, to study the effect of inertia on Turing instabilities [205, 206]. This is our preferred approach to describe reaction-transport systems with inertia, since it has a solid mesoscopic foundation. We consider Turing instabilities for two classes of reaction random walks, DIRWs and DDRWs, see Sect. 5.6. 

---