Anomalous Diffusion With Reactions And Non-ergodic Dynamics:Models,theory And Applications

As a rich research field highly related to physics and mathematics,anomalous diffusion and the associated fractional-order equations have recently attracted wide interests of researchers from different fields.This dissertation is devoted to explore three important issues within the field of anomalous diffusion:the ergodicity of biased continuous time random walk models,the Feynman-Kac equations of reaction diffusion processes,and the generalized Fokker-Planck equations with absorptions under the harmonic potential.Totally five chapters constitute this dissertation.Chapter 1 elaborately illustrates the research background of anomalous diffusion,the work of the dissertation and the preliminary knowledge.Chapter 2 examines the biased continuous time random walk?CTRW?models with power-law waiting time distributions?WTDs?via computing analytically their ensemble and time averaged spreading variance.A new definition of ergodicity is proposed for this non-zero drift case.All possible values of the scaling exponentare considered for the WTD???1/¹⁺,which exhibit different characteristics.Specifically,when 0<<1,the mean waiting time diverges and the CTRW process is non-ergodic in the sense of the non-equivalence of the ensemble and time averaged displacement characteristics.When>1,the mean waiting time is finite and two kinds of renewal processes?namely,both ordinary and equilibrium CTRWs?are investigated.CTRW processes with 1<<2 are ergodic for the equilibrium and non-ergodic for the ordinary situation.Furthermore,CTRW processes with>2?both for the equilibrium and ordinary situation?are always ergodic.For biased CTRWs with>1 the behaviour of the ergodicity breaking parameter is also investigated.In addition,we demonstrate that for biased CTRWs the Einstein relation is valid on the level of the ensemble and time averaged displacements,in the entire range of the WTD exponent.Chapter 3 provides a theoretical framework for deriving the forward and back-ward Feynman-Kac equations for the distribution of functionals of the path of a par-ticle undergoing both diffusion and reaction processes.Once given the diffusion type and reaction rate,a specific forward or backward Feynman–Kac equation can be obtained.The results in this chapter include those for normal/anomalous diffusions and reactions with linear/nonlinear rates.Using the derived backward Feynman-Kac equations,we apply our findings to compute the distributions of some physical statistics,including the occupation time in half-space,the first passage time,and the occupation time in half-interval with an absorbing or reflecting boundary,for the physical system with anomalous diffusion and spontaneous evanescence.Chapter 4 considers the Fokker-Planck equations with reactions,as a general-ization of the classic Fokker-Planck equations.Firstly the reaction diffusion equation without external potential is investigated via computing analytically its solution and survival probability.Moreover,based on the solution of the classic Fokker-Planck equation in the presence of an attractive harmonic potential,the properties of the classic Fokker-Planck equation with reactions?namely,the reaction diffusion equa-tion with a harmonic potential?are validated and discussed,such as the survival probability and the scaling limit when the absorptions are localized at 0.Last but not least,the fractional Fokker-Planck equation with reactions is presented and its connections to the classic situation are carefully derived,especially regarding the survival probability.Chapter 5 summarizes the dissertation and presents some outlooks of several interesting research projects in this field.