Non-Markovian CTRW Models with Chemical Reactions

4 Non-Markovian CTRW Models with Chemical Reactions 

In this section we consider CTRW models for which the waiting time distribution is not exponential. The main challenge is to incorporate nonlinear kinetic terms into non-Markovian transport equations. Several approaches exist in the literature about how to include kinetic terms in reaction-transport systems with anomalous diffusion. We discuss them in detail in the following. 

We consider a one-component reaction-transport system consisting of particles that follow a CTRW. Let p x, t) represent the density of these particles at point X and time t. We write the reaction term in the form F(p) = f(p)p. It is also convenient to represent the nonlinear reaction rate f p) as the difference between 

3 Random Walks and Mesoscopic Reaction-Transport Equations 

Since the kinetics is of the form / (p)p, the change in the number of particles, of a given type, between the jumps involves the exponential factor exp[/ / p x, u))du]. To explain this, let us introduce the density of particles j (x, r) arriving at point x exactly at time r. During time interval (t, t), this density changes as follows  

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