Random Models,Analyses In Non-Ergodic Diffusion And Monte Carlo Simulations

This paper mainly discusses about the theory of nonergodic anomalous diffusion,each theoretical result is also verified by Monte Carlo numerical method.When u-tilizing random walk theory to analyze nonergodic anomalous diffusion,we always construct two different kinds of independently identically distributed random vari-ables,i.e.,waiting time and jump length.However,the random walk particle may have different kinds of distributions of waiting time or jump length when it moves to a different area.In order to solve this problem,we introduce a concept of internal state and generalize the classical random walk theory.On the other hand,this paper also systematically study on random walk with coupled space and time,we generalize the method of coupling from linearity to a general way.Then we calculate some statistical quantities by utilizing orthogonal polynomials,and further this method can also be used to solve the random walk with coupled space and time in a Harmonic potential theoretically.This paper consists of seven chapters.In the first chapter,we briefly introduce the history and physical background of fractional differential equation and noner-godic anomalous diffusion,and analyze the current researches.Then we give a brief description of the research content,method and the creative points of this paper.Second chapter mainly studies on the compound Poisson process with multi-ple internal states.First we introduce the classical continuous time random walk model,which can be considered as one kind of compound Poisson process.Then we introduce the concept of internal state into continuous time random walk model,some introductions of the related background are also made to show the motivation of multiple internal states.Then we will continue to derive the macroscopic equa-tion for the probability density function of the position of particle at some time,which is Fokker-Planck equation,then through constructing some transit matrices and waiting time,we obtain the asymptotic behaviors of the second moment,besides we analyze the transit trend of diffusion exponent.After defining the functionals of trajectories of particles or the internal state,the corresponding macroscopic equation of probability density function of each functional,which is known as Feynman-Kac equation,is also derived,respectively.Some concrete examples and applications for these two Feynman-Kac equations are also given.In the final part of this chapter,we will utilize the compound Poisson process with multiple internal states to deal with the non-immediately-repeating process,and the second moment is used to reflect the speed of diffusionIn the third chapter,basing on the continuous time random walk model,we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents,often observed in the natural world.In the continuous time random walk framework,we take the waiting time probability density function as an infinite series in three parameter Mittag-Leffler functions.According to the models,the mean squared displacement of the process is analytically obtained and numerically verified,in particular,the trend of its transition is shown;furthermore the stochastic representation of the macroscopic equation is also presented.Finally,the fractional moments of the model are calculated,and the analytical solutions of the model with external harmonic potential are obtained and some applications are proposed.In the forth chapter,we will transfer our problem from the random walk theory with independent space and time to the coupled one.The random walk with coupled space and time,which is also known as Levy walk,has plenty of applications in mathematics and physics.At first,we will introduce the basic theory and motivations of Levy walks.Then we will construct the model of Levy walk with multiple internal states,and obtain the form of probability density function after applying Fourier and Laplace transforms on space and time variables respectively.We will also consider the non-immediately-repeating Levy walk as an application,we discover for the Levy walk belonging to super diffusion region,non-immediately-repeating condition has no influence on its Person constant and mean squared displacement,this is kind of stable property for Levy walk.However,if the Levy walk shows the dynamical behaviors of normal diffusion,then the effect of non-immediately-repeating emerges.For the Levy walk with some particular transit matrices,it may display nonsymmetric dynamics in these cases,the behaviors of their variances are discussed in detail,especially some comparisons with the ones of the continuous time random walks are made(a striking difference is the changes of the exponents of the variances).The first passage time distribution and its average of Levy walks are simulated,the results of which turn out that the first passage time can distinguish Levy walks with different transit matrices,while the MSD cannotIn chapter five,we will deal with the problem of Levy walk with parameter de-pendent velocity by utilizing Hermite polynomials.To analyze stochastic processes,one often uses integral transform(including Fourier and Laplace transforms)meth-ods.However,for the time-space coupled cases,e.g.the Levy walks,sometimes the integral transform method may fail.In this chapter,we provide a Hermite polyno-mial expansion approach,being complementary to the integral transform method,to the Levy walks.Two approaches are compared for some already known results.We also consider the Levy walk with parameter dependent velocity,namely,we consider the Levy walk with velocity which depends on the walking length or the duration of each step.Some interesting features of the generalized Levy walks are observed,including the special shapes of the probability density function,the first passage time distributions,and various diffusive behaviors of the mean squared displacementIn chapter six,we will discuss the influence of harmonic potential on Levy walks At first we will expand the probability density function of Levy walks undergoing harmonic potential by utilizing Hermite polynomials,and some statistical quantities and the approximate form of stationary distribution by truncating some terms are also calculated.Meanwhile under the influence of harmonic potential,the approximated form of stationary distribution of Levy walk with reflecting boundary condition at the origin is also obtained.Our results settle some longstanding puzzles around Levy walks,and it also turns out that the Hermite expansion has much more potentials in dealing with problems of Levy walksIn the seventh chapter,we will conclude the whole paper and make expectation of works in the future.