Fluctuation Theory And Simulation For Nonergodic Diffusion

The purpose of this research project is to investigate the difference between rare fluctuations and typical fluctuations,and their relations,in which the big jump principle is introduced to connect the rare events and the single big waiting time.Besides,we build the models governing the corresponding functional distributions,namely the forward and backward aging Feynman-Kac equations.More details are as follows:In the first chapter of the thesis,we give the reasons why this research project is interesting and important.The introduction of the background is presented,which provides the reader with essential context needed to understand the research problem and its significance.In addition,the five related models are described,namely renewal process,L?evy flight,continuous time random walk,aging continuous time random walk,and L?evy walk.Renewal processes,studied in the second chapter,are simple stochastic models for events that occur on the time axis when the time intervals between events are independent and identically distributed random variables.This idealized approach has many applications,ranging from the analysis of photon arrival times to queuing theory.To describe rare events,the rate function approach from large deviation theory does not hold and new tools must be considered.Here we investigate the large deviations of the number of renewals,the forward and backward recurrence time,the time interval straddling the observation time,and the occupation time.We show how non-normalized densities describe these rare fluctuations,and how moments of certain observables are obtained from these limiting laws.Numerical simulations illustrate our results showing the deviations from arcsine,Dynkin,Darling-Kac,L?evy,and Lamperti limits laws.We further turn our attention to the biases continuous time random walk.Continuoustime random walk describes anomalous dynamics charactered by a series of jumps separated by random waiting times.Using an analytical approach,we derive the rare fluctuations which,described by non-normalized density,while in this non-normalized state yields the high order moments.We further obtain the typical fluctuations probability density function,ranging from Gaussian to L?evy statistics,which extends the classical L?evy law.Furthermore,we show how the Fokker-Planck equations describe the typical fluctuations,and how the parameters change the solution of the Fokker-Planck equation.Utilizing the big jump principle,we construct a relation between the rare events and the single big waiting times.In the fourth and fifth chapters,the models governing the corresponding functional distributions are analyzed,and the corresponding aging forward and backward FeynmanKac equations are obtained.We further discuss a more complex situation,in which prescribed function U depend on the position of the particles and the observation time.Besides,we also discuss the effect of the force on the functional distributions.At the end of each chapter are a series of conclusions,including the theoretical results and simulations.In chapter six,the results are summarized and the possible further research works are indicated.All the theoretical results are confirmed by numerical simulations.