| Transport processes,especially diffusion processes,are ubiquitous in both physi-cal and biological systems.Brownian motion(normal diffusion)has been playing the dominating role in the field of diffusion for long times.In recent decades,however,non-Brownian motion(anomalous diffusion)was found in great quantities in ex-periments,and increasingly shook the absolute dominance of the Brownian motion.Brownian motion has the property of ergodicity which implies that the time average is equal to the ensemble average.However,a large number of anomalous diffusion processes are found to be non-ergodic,different from Brownian motion.Further,in studying the intrinsic mechanism of anomalous and non-ergodic diffusion,it is found that the motion of particles is not as simple as described by Brownian motion.In stead,they are often influenced by heterogeneous environments or their motion patterns are inhomogeneous(having different states or phases).In this dissertation,we will discuss the anomalous diffusion and non-ergodic behavior of these stochastic processes and their applications,such as Feynman-Kac equation,occupation time and first-passage time.The specific contents of this dissertation are as follows:In chapter one,we briefly introduce the research background of anomalous diffusion and non-ergodic behavior.Then we introduce the structure of this paper and briefly analyze the contents of each chapter from the perspective of research object,research method and innovation.Finally,for the convenience of discussing the problems in this dissertation,we make a certain degree of elaboration on some basic anomalous diffusion models and research methods involved in this dissertation.In chapter two,we describe the heterogenous diffusive process under the effect of an external force by an overdamped Langevin equation,and derive the corresponding Feynman-Kac equation,which governs the probability density function(PDF)of the path functional.The functionals have diverse applications in physics,mathematics,hydrology,economics,and other fields.Under the framework of a continuous-time random walk,the Feynman-Kac equations,including those of the paths of stochastic processes of normal diffusion,anomalous diffusion,and even diffusion with reaction,have been derived.Sometimes the stochastic processes in physics and chemistry are naturally described by Langevin equations.The Langevin picture has the significant advantage of studying the dynamics with an external force field and analyzing the effect of noise resulting from a fluctuating environment.Based on the Langevin equation and the technique of Fourier transform,we derive the forward and backward Feynman-Kac equations.For the newly built equations,their applications in solving the PDFs of the occupation time and area under the trajectory curve are provided,and the results are confirmed by simulations.In chapter three,we still consider the heterogeneous diffusion process affected by external forces in last chapter,but focus on the ergodic and non-ergodic behavior of the process.Since our main concern is the large-x behavior for long times,the diffusivity and potential are,respectively,assumed as the power-law forms D(x)=D0|x|α and U(x)=U0|x|β for simplicity.Based on the competition roles played by D(x)and U(x),three different cases,β>α,β=α,and β<α,are discussed.The system is ergodic for the first case β>α,where the time average agrees with the ensemble average.By contrast,the system is non-ergodic for β<α,where the relation between time average and ensemble average is uncovered by infinite-ergodic theory.For the middle case β=α,the ergodic property depends on the prefactors D0 and U0,becoming more delicate.The PDFs of the time averaged occupation time for three different cases are also evaluated and confirmed by simulations.In chapter four,we mainly discuss the anomalous and non-ergodic behaviors of a class of inhomogeneous two-state stochastic processes.With the rich dynamics studies of single-state processes,the two-state processes are attracting more interest,since they are widely observed in complex system and have effective applications in diverse fields,such as foraging behavior of animals.This chapter builds the theo-retical foundation of the process with two states:Levy walk and Brownian motion,which have been proved to be an efficient intermittent search process.The sojourn time distributions in two states are both assumed to be heavy-tailed with expo-nents α±∈(0,2).The dynamical behaviors of this two-state process are obtained through analyzing the ensemble-averaged and time-averaged mean-squared displace-ments(MSDs)in weak and strong aging cases.It is discovered that the magnitude relationship of α±decides the fraction of two states for long times,playing a crucial role in these MSDs.According to the generic expressions of MSDs,some inherent characteristics of the two-state process are detected and discussed in detail for six different cases.In chapter five,we continue to discuss the inhomogeneous two-state processes of the previous chapter,but focus on their strong anomalous diffusion behaviors,which are often observed in complex physical and biological systems,and characterized by the nonlinear spectrum of exponents qv(q)by measuring the absolute q-th moment(|x|q>.We still assume that the two states are Levy walk and Brownian motion with their sojourn times obeying the power law distributions with exponents α±∈(0,2).Detailed scaling analyzes are performed for the coexistence of three kinds of scalings in this system.Different from the pure Levy walk,the phenomenon of strong anomalous diffusion can be observed for this two-state process even when the exponent of power law distribution at Levy walk phase satisfies α+<1.When 1<α+<2,the PDF in the central part becomes a combination of stretched Levy distribution and Gaussian distribution due to the long sojourn time in Brownian phase,while the PDF in the tail part(in the ballistic scaling)is still dominated by the infinite density of Levy walk.In chapter six,we make a brief summary of the research contents and main conclusions of this dissertation.A prospect of the problems to be studied after graduation are provided finally. |
