CTRW Models and Front Propagation

In this section we present a geometric approach to the problem of propagating fronts into an unstable state, valid for an arbitrary CTRW with a KPP reaction rate. We derive an integral Hamilton-Jacobi type equation determining the position of the front and its velocity. The essential feature of the method is that it does not rely on the explicit derivation of an evolution equation for the particle density. In particular. 

In previous sections we showed that the macroscopic dynamics of propagating fronts depend on the statistical characteristics of the underlying random walk model for the mesoscopic transport process. Since the dynamics of fronts are nonuniversal, it is an important problem to find universal rules relating both levels of description. The goal of this section is to address this problem. We are interested in exploring the physical properties of systems of particles that disperse according to a general CTRW. 

As usual we introduce the mesoscopic concentration /o(x, t) of particles performing a CTRW. The complete description of the mesoscopic transport processes is given by the joint probability density V (t, z) of making a jump of length z in the time interval (r, t + dr), see Sect. 3.2. We assume that the local growth rate has the 

We start with Model C given by (3.141) and the above kinetics. The governing equation for p(x, t) can be written in the form 

This equation describes the balance of particles at the position x at time t. The first term on the RHS of (5.27) represents the number of particles remaining at their initial position x up to time t. The second term corresponds to the number of particles arriving at x up to time t from position x — z and time t — x, and the last term is a production term due to growth (5.26). 

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