Continuous time random walk microscopic models

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

It is a vector-matrix equation in s space and an integral equation in time. The unknown is the vector of functions This equation is called continuous time random walk (CTRW) and was used in phenomenological modeling of transport [17]. Equation 13.1 is closed and can be solved provided that A. /x) is known. Our contribution is to show how detailed microscopic dynamics is used to compute or its moments (see below). 

A variety of models exist to explain these similarities on a microscopic level. The simplest of them is the random free energy barrier model developed by Dyre (1988). In this model hopping is assumed to be basic mechanism for conducticHi where hopping takes place over spatially varying energy barriers. Within the continuous time random walk approximation (MontroU and Weiss 1965), this model results in... 

This is best understood intially by considering the process of diffusion. Ghromatographic peaks represent chemical species that have been concentrated in space and time and the process of diffusion will immediately disperse them in space as a function of time. The conceptual basis of diffusion lies in the concept of the random walk model, wherein particles/molecules in suspension or solution are being jostled continuously by collisions with other particles or molecules. This is also referred to as Brownian motion, and is readily apparent when observing small particles with a microscope, such as some pollen grains, that seem to be in constant and random motion as they gradually spread out from any center of concentration. 

So far we have considered a single mesoscopic equation for the particle density and a corresponding random walk model, a Markov process with continuous states in discrete time. It is natural to extend this analysis to a system of mesoscopic equations for the densities of particles Pi(x,n), i = 1,2,..., m. To describe the microscopic movement of particles we need a vector process (X , S ), where X is the position of the particle at time n and S its state at time n. S is a sequence of random variables taking one of m possible values at time n. One can introduce the probability density Pj(jc, n) = 9P(X < x,S = i)/dx and an imbedded Markov chain with the m x m transition matrix H = (/i ), so that the matrix entry corresponds to the conditional probability of a transition from state i to state j. 

---