Anomalous Diffusion Processes And Their Fractional Kinetic Equations: Modeling By Coupled Continuous Time Random Walk Models And Subordination

In recent years, many researchers have been attracted by anomalous diffusion phenomena. There exist three kinds of models describing anomalous diffusion processes. Among them continuous time random walk(CTRW) is the most intuitional and has the widest applications. In this thesis, based on continuous time random walk theory, we discuss statistical features of anomalous diffusion by generalized master equation and subordination technique. Since the disadvantage in the CTRW models is that there is no straightforward way to incorporate external force ?elds,the alternative approach is given in terms of fractional equation which appear to be tailored for such kind of problem like the consideration of external force ?elds.Therefore, fractional kinetic equations describing anomalous kinetics are also hot topics. Here, we arrange relevant discussions and detailed work as follows:In chapter 3, ?rstly, we connect generalized Kolmogorov-Feller equation with decoupled CTRW models using Montroll-Weiss equation. Secondly, we consider a coupled CTRW model with jump length dependent on waiting time. By suitable settings for waiting time probability density function(PDF) and jump length conditional PDF, we obtained the asymptotic forms of jump PDF in Fourier-Laplace domain. At last, substituting the asymptotic forms of jump PDF in Fourier-Laplace domain and those of waiting time PDF in Laplace domain into Montroll-Weiss equation, we obtained the algebraic relation which the PDF of coupled CTRW processes obey in Fourier-Laplace domain. By suitable deformation, and then taking inverse Fourier-Laplace transform for algebraic equation, we obtained fractional diffusion equation which the PDF of coupled CTRW processes satisfy in space-time domain.We also computed mean square displacement(MSD) of the introduced model and used MSD to discriminate the type of anomalous diffusion.In chapter 4, similar discussions as chapter 3, we consider a special random walk: a directed coupled CTRW. For this situation, the Fourier-Laplace transform of jump PDF in Montroll-Weiss equation should be changed into the Laplace-Laplace transform. Assumed that jump length conditional PDF is Dirac δ- function and waiting time PDF has long tail distribution, we derived a combined fractional drift equation that the proposed model’s PDF obeys.In chapter 5, using subordination concept, we decompose the decoupled CTRW process into two stochastic processes: discrete time random walk process and counting process. Replacing internal variable ”the number of steps” by random variable, one gets an anomalous diffusion process. The internal random variable is named as random time. Here, we introduce three types of random time processes and discuss their properties. The common structure of the proposed random time processes consists of power function of time variable and negative power function of an α-stable random variable. And then, we discuss the statistic properties of anomalous diffusion processes with the proposed three types of random time and also derive corresponding fractional Fokker-Planck-Type equation.