Analyses And Monte Carlo Simulations In Non-ergodic Anomalous Dynamic Model In Complex Environment

In addition to normal diffusion,anomalous diffusions are ubiquitous in all areas of the natural world.More and more scholars aim to study the dynamic behaviours of anomalous diffusion processes since its universality.In this paper we analyze the transport properties of the anomalous diffusion processes in complex environment based on the random walk theory.We utilize the Hermite orthogonal polynomial approximation to deal with the problem of time-space coupling in complex environment.In addition,the uniform expansion or contraction of the medium induces a displacement for diffusion particle in one dimension uniform non-static media.It is difficult to directly analyze the transport properties of diffusion particle in this media since the distribution of this displacement is unknown.We introduce the comoving coordinate to solve this kind of problem.Finally,based on the generalized diffusion equation we derive the search reliability and search efficiency for different types of memory kernel which helps one to formulate the optimal search strategy for different complex environments.All the theoretical results are verified by Monte Carlo method.This paper consists of seven chapters.In the first chapter,we briefly introduce the research background and significance of anomalous diffusion process,also including the continuous time random walk model and L′evy walk model as well as their Monte Carlo simulation algorithm codes.Then we give a brief description about the main research content of this paper,the research methods and the innovation points of each chapter.In the second chapter,we focus on the L′evy walk model in external potential field.We first study the dynamics of L′evy walks in the linear potential field by building the master equation which governing the probability density function of the position of the L′evy walk particles under the action of linear potential.Employing Hermite polynomial approximation to deal with the challenge of spatiotemporally coupled,some transport properties are observed.Thus,we can observe the influence of linear potential on the L′evy walk process.Next,we research the model of L′evy walks in linear potential combined with harmonic potential field.Some prominent features are observed,such as strongly anomalous diffusion,non-Gaussian distribution,and stationary diffusion,etc.In addition,the relaxation dynamics of L′evy walk in mixed potential are discussed.In the third chapter,a stochastic process including three different phases is considered.These three phases respectively are movement phase,return phase,and rest phase.The particles move according to Brownian motion or the ballistic type of L′evy walk in the movement phase.In the return phase,the particles start from the finally position of the movement phase and back to the origin in three different ways: with a constant velocity or constant acceleration or under the harmonic potential.After return to the origin,the particles rest at the origin with a random time which obeys the rest time probability density function.The long-time asymptotic expression of the mean squared displacements corresponding to the diffusion process under different movement dynamics,returning,and resting time distributions are derived.We further consider the stationary distribution when the mean squared displacement is localized.In addition,the mean first passage time is calculated when the particles move following Brownian motion in the movement phase.In the forth chapter,we concentrate on studying the dynamic of L′evy walk model in one-dimension uniform non-static media.We establish the master equation in comoving coordinate since the physical coordinate relates to comoving coordinate by scale factor.Some statistics corresponding to the diffusion process in both coordinates are obtained by utilizing the Hermite orthogonal polynomial approximation,such as mean squared displacements and kurtosis.The mean first passage time of L′evy walk in comoving coordinate is also considered through Monte Carlo simulations method.In the fifth chapter,we focus on studying L′evy walk dynamics in an external linear potential field in non-static media and discussing the combined action of nonstatic media and linear potential to L′evy walk model.Given some representative scale factor and walking time distribution of L′evy walk,employing the Hermite orthogonal polynomial approximation to deal with the master equation in comoving coordinate,we observe some striking and interesting results which stem from the interplay between the determined motion induced by the non-static media,the motion caused by the linear potential and intrinsic motion of L′evy walk.In the sixth chapter,we consider one-dimensional Brownian search as well as L′evy search in presence of trapping.Based on the Montroll-Weiss equation of continuous time random walk,the generalized diffusion equation of the search diffusion are derived by introducing the memory kernel function to represent the waiting time distribution.The first arrival time density,search reliability as well as search efficiency are obtained for some representative search processes which are determined by the memory kernel.These results further help one to develop the best search strategy in different complex environments.In addition,we extend the above method to the case of multiple targets as well as combined search strategies.In the seventh chapter,we summarize the paper and outlook the future research work.