Langevin Descriptions And Analyses Of The Anomalous Diffusion Processes In External Force Fields

The diffusion phenomenon is very common and deserved to be studied in many physical and biological systems.At the beginning,Brown observed that pollen particles on liquid can exhibit disorganized,irregular movement.This phenomenon is called the normal diffusion phenomenon.Then,more and more anomalous diffusion processes are observed and researched.But most research works are focusing on the motion of the free particles,i.e.,the particles are not affected by the external forces.However,most of the particles in nature are affected by the external forces.These external forces may depend on time,or depend on the position,or remain unchanged,and so on.This dissertation pays attention to the influence of the external forces on the anomalous diffusion processes,including the subdiffusion process and superdiffusion process.We study the type of the diffusion and the properties of the particles in external potentials,and derive the corresponding Fokker-Planck equation.Through the calculation of various statistics and the derivation of macroscopic diffusion equations,we can analyze the influence of external force fields on stochastic processes in detail.In addition,we also pay attention to the resonance phenomenon caused by the external periodic force.The specific contents of this dissertation are as follows:In chapter one,we mainly introduce the background knowledge about the diffusion of particles,including the behavior of the diffusion,the models which can describe the corresponding stochastic processes,the properties of the particles as well as the corresponding statistical quantities.This will pave the way for further research in the following chapters.In addition,we introduce some special stochastic processes – subordination process and its inverse process.They can be regarded as a kind of time-changed stochastic process,which is an important part of the theoretical basis of this dissertation.We also introduce several methods commonly used in this dissertation,such as Laplace and Fourier transforms,and list their common properties.In the end,we present the objects,methods and innovation of our work as well as the connection between the following chapters.In chapter two,to investigate the effects of external potentials,say,harmonic potential,linear potential,and time-dependent force,we study the subdiffusion described by the subordinated Langevin equation with white Gaussian noise or,equivalently,by the single Langevin equation with compound noise.The reason for subdiffusion phenomenon is that the particles are trapped somewhere for long times,so there are two ways of external force acting on subdiffusion stochastic process.One is that the external forces continuously act on the subdiffusion process.The other is that the external forces intermittently act on the subdiffusion process,that is,they only act on the subdiffusion process when the particle jumps.When the particle is trapped somewhere and cannot move,the external forces do not act.Some common statistical quantities,such as the mean squared displacements,position autocorrelation function,and generalized Einstein relation,are discussed to distinguish the effects of various forces and different patterns of acting.It is found that for the first acting pattern of the force,if the external force does not depend on the position of the particle,it will not change the original diffusion behavior,ergodic property and correlation coefficient of the particle;However,in the same pattern of acting,the external force depending on the position will change these properties of the particle.For the second acting pattern of the force,the position-independent and position-dependent external forces will affect most of the statistics of the stochastic process.In addition,we also derive the corresponding Fokker-Planck equations.Different acting patterns of the external forces lead to different fractional derivative operators of the Fokker-Planck equation.In chapter three,we establish the Langevin equation coupled with a subordinator to study the influence of external constant force,time-dependent periodic force and linear force on L?evy walk.L?evy walk exhibits superdiffusion phenomenon and keep moving,so here we focus on the case where the external forces continually affect the stochastic process.Based on the established underdamped Langevin equations,we can calculate the statistics of stochastic process in various external force fields,such as velocity correlation function,mean squared displacements.More specifically,for the case of constant force,the external force field accelerates the diffusion of particles.Besides,the superballistic diffusion phenomenon and the non-ergodic behavior is always presented in the L?evy walk process.For the case of time-dependent periodic force,the external force field only affects the coefficient of the mean squared displacement of the L?evy walk,and the “ultraweak” non-ergodic behavior of the free L?evy walk is retained,not affected by the external force field.Besides,the corresponding generalized Klein-Kramers equation is obtained.The method to derive this equation is also valid for other types of external forces.For the case of harmonic potential(linear force),we derive the general formula of the mean squared displacement of stochastic processes in harmonic potential described by the overdamped Langevin equation.We find that as the time goes on,ensemble-averaged mean squared displacement of the confined L?evy walk affected by the harmonic potential will eventually tend to a stable value.In addition,we also study its nonergodic behavior.In this chapter,Monte Carlo method is used to simulate various statistics to verify the correctness of the theoretical results.In chapter four,we analyzes the anomalous resonant behaviors of the generalized Langevin system with a multiplicative dichotomous noise and an internal tempered Mittag-Leffler noise.This Langevin equation satisfies the fluctuationdissipation theorem.We know that noise is not only the origin of uncertainty but also plays a positive role in helping to detect signals with information.For a system with a fluctuating harmonic potential,by use of the Shapiro-Loginov formula,we obtain some exact expressions to reveal the stochastic resonance,such as the first moment,the amplitude and autocorrelation function for the output signal as well as the signal-noise ratio.The phenomenon of stochastic resonance we study here refers to the generalized stochastic resonance,that is,the output signal or some functions of the output signal(such as moments,autocorrelation function,signal-noise ratio)show non-monotonic behavior with the change of noise and system parameters.We try to reveal how the amplitude of the output signal depends on the system parameters and analyze the effect of exponential truncation on the behavior of stochastic resonance.We also pay attention to the dependence of signal-noise ratio on system parameters and the influence of the tempering parameter.In order to obtain theoretical results,we have carried out detailed analysis,including the influence of tempering parameter and memory exponent on stochastic resonance,and how the critical memory exponent changes with the increase of tempering parameter,and so on.In addition,in order to show the conclusions of resonance more intuitively,we exhibit the non-monotonic dependence between the amplitude of the output signal as well as the signal-noise ratio and system parameters through a large number of images.Finally,almost all the theoretical results are verified by the numerical simulations.In chapter five,we make a conclusion of our research works and make a prospect of the subjects to be studied.