arXiv subjects

Weihua Deng

Publications and source records attributed to Weihua Deng.

At least 55 records · Page 3Linked to original sources

arXiv2019

Langevin picture of Lévy walk in a constant force field

Lévy walk is a practical model and has wide applications in various fields. Here we focus on the effect of an external constant force on the Lévy walk with the exponent of the power-law distributed flight time $α\in(0,2)$. We add the term $Fη(s)$ ($η(s)$ is the Lévy noise) on a subordinated Langevin system to characterize such a constant force, being effective on the velocity process for all physical time after the subordination. We clearly show the effect of the constant force $F$ on this Langevin system and find this system is like the continuous limit of the collision model. The first moments of velocity processes for these two models are consistent. In particular, based on the velocity correlation function derived from our subordinated Langevin equation, we investigate more interesting statistical quantities, such as the ensemble- and time-averaged mean squared displacements. Under the influence of constant force, the diffusion of particles becomes faster. Finally, the super-ballistic diffusion and the non-ergodic behavior are verified by the simulations with different $α$.

cond-mat.stat-mech↗

arXiv2019

Continuous time random walks and Lévy walks with stochastic resetting

Intermittent stochastic processes appear in a wide field, such as chemistry, biology, ecology, and computer science. This paper builds up the theory of intermittent continuous time random walk (CTRW) and Lévy walk, in which the particles are stochastically reset to a given position with a resetting rate $r$. The mean squared displacements of the CTRW and Lévy walks with stochastic resetting are calculated, uncovering that the stochastic resetting always makes the CTRW process localized and Lévy walk diffuse slower. The asymptotic behaviors of the probability density function of Lévy walk with stochastic resetting are carefully analyzed under different scales of $x$, and a striking influence of stochastic resetting is observed.

cond-mat.stat-mech↗

arXiv2019

Lévy walk revisited: Hermite polynomial expansion approach

Integral transform method (Fourier or Laplace transform, etc) is more often effective to do the theoretical analysis for the stochastic processes. However, for the time-space coupled cases, e.g., Lévy walk or nonlinear cases, integral transform method may fail to be so strong or even do not work again. Here we provide Hermite polynomial expansion approach, being complementary to integral transform method. Some statistical observables of general Lévy walks are calculated by the Hermite polynomial expansion approach, and the comparisons are made when both the integral transform method and the newly introduced approach work well.

cond-mat.stat-mech↗

arXiv2019

Theory of relaxation dynamics for anomalous diffusion processes in harmonic potential

Optical tweezers setup is often used to probe the motion of individual tracer particle, which promotes the study of relaxation dynamics of a generic process confined in a harmonic potential. We uncover the dependence of ensemble- and time-averaged mean square displacements of confined processes on the velocity correlation function $C(t,t+τ)$ of the original process. With two different scaling forms of $C(t,t+τ)$ for small $τ$ and large $τ$, the stationary value and the relaxation behaviors can be obtained immediately. The gotten results are valid for a large amount of anomalous diffusion processes, including fractional Brownian motion, scaled Brownian motion, and the multi-scale Lévy walk with different exponents of running time distribution.

cond-mat.stat-mech↗

arXiv2019

Numerical algorithm for the space-time fractional Fokker-Planck system with two internal states

The fractional Fokker-Planck system with multiple internal states is derived in [Xu and Deng, Math. Model. Nat. Phenom., $\mathbf{13}$, 10 (2018)], where the space derivative is Laplace operator. If the jump length distribution of the particles is power law instead of Gaussian, the space derivative should be replaced with fractional Laplacian. This paper focuses on solving the two state Fokker-Planck system with fractional Laplacian. We first provide a priori estimate for this system under different regularity assumptions on the initial data. Then we use $L_1$ scheme to discretize the time fractional derivatives and finite element method to approximate the fractional Laplacian operators. Furthermore, we give the error estimates for the space semidiscrete and fully discrete schemes without any assumption on regularity of solutions. Finally, the effectiveness of the designed scheme is verified by numerical experiments.

math.NA↗

arXiv2019

Aging two-state process with Lévy walk and Brownian motion

With the rich dynamics studies of single-state processes, the two-state processes attract more and more interests of people, since they are widely observed in complex system and have effective applications in diverse fields, say, foraging behavior of animals. This report builds the theoretical foundation of the process with two states: Lévy walk and Brownian motion, having been proved to be an efficient intermittent search process. The sojourn time distributions in two states are both assumed to be heavy-tailed with exponents $α_\pm\in(0,2)$. The dynamical behaviors of this two-state process are obtained through analyzing the ensemble-averaged and time-averaged mean squared displacements (MSDs) in weak and strong aging cases. It is discovered that the magnitude relationship of $α_\pm$ 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. The effects of the fraction on these observables are detailedly presented in six different cases. The key of getting these results is to calculate the velocity correlation function of the two-state process, the techniques of which can be generalized to other multi-state processes.

physics.class-ph↗

arXiv2019

Subdiffusion in an external force field

The phenomena of subdiffusion are widely observed in physical and biological systems. To investigate the effects of external potentials, say, harmonic potential, linear potential, and time dependent force, we study the subdiffusion described by subordinated Langevin equation with white Gaussian noise, or equivalently, by the single Langevin equation with compound noise. If the force acts on the subordinated process, it keeps working all the time; otherwise, the force just exerts an influence on the system at the moments of jump. Some common statistical quantities, such as, the ensemble and time averaged mean squared displacement, position autocorrelation function, correlation coefficient, generalized Einstein relation, are discussed to distinguish the effects of various forces and different patterns of acting. The corresponding Fokker-Planck equations are also presented. All the stochastic processes discussed here are non-stationary, non-ergodicity, and aging.

cond-mat.stat-mech↗

arXiv2019

Ergodic properties of heterogeneous diffusion processes in a potential well

Heterogeneous diffusion processes can be well described by an overdamped Langevin equation with space-dependent diffusivity $D(x)$. We investigate the ergodic and non-ergodic behavior of these processes in an arbitrary potential well $U(x)$ in terms of the observable---occupation time. Since our main concern is the large-$x$ behavior for long times, the diffusivity and potential are, respectively, assumed as the power-law forms $D(x)=D_0|x|^α$ and $U(x)=U_0|x|^β$ 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, being both determined by the steady solution for long times. In 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, depending on the prefactors $D_0$ and $U_0$, becomes more delicate. The probability density distribution of the time averaged occupation time for three different cases are also evaluated from Monte Carlo simulations.

cond-mat.stat-mech↗

arXiv2019

Fast algorithms for convolution quadrature of Riemann-Liouville fractional derivative

Recently, the numerical schemes of the Fokker-Planck equations describing anomalous diffusion with two internal states have been proposed in [Nie, Sun and Deng, arXiv: 1811.04723], which use convolution quadrature to approximate the Riemann-Liouville fractional derivative; and the schemes need huge storage and computational cost because of the non-locality of fractional derivative and the large scale of the system. This paper first provides the fast algorithms for computing the Riemann-Liouville derivative based on convolution quadrature with the generating function given by the backward Euler and second-order backward difference methods; the algorithms don't require the assumption of the regularity of the solution in time, while the computation time and the total memory requirement are greatly reduced. Then we apply the fast algorithms to solve the homogeneous fractional Fokker-Planck equations with two internal states for nonsmooth data and get the first- and second-order accuracy in time. Lastly, numerical examples are presented to verify the convergence and the effectiveness of the fast algorithms.

math.NA↗

arXiv2019

First exit and Dirichlet problem for the nonisotropic tempered $α$-stable processes

This paper discusses the first exit and Dirichlet problems of the nonisotropic tempered $α$-stable process $X_t$. The upper bounds of all moments of the first exit position $\left|X_{τ_D}\right|$ and the first exit time $τ_D$ are firstly obtained. It is found that the probability density function of $\left|X_{τ_D}\right|$ or $τ_D$ exponentially decays with the increase of $\left|X_{τ_D}\right|$ or $τ_D$, and $\mathrm{E}\left[τ_D\right]\sim \left|\mathrm{E}\left[X_{τ_D}\right]\right|$,\ $\mathrm{E}\left[τ_D\right]\sim\mathrm{E}\left[\left|X_{τ_D}-\mathrm{E}\left[X_{τ_D}\right]\right|^2\right] $. Since $\mathrmΔ^{α/2,λ}_m$ is the infinitesimal generator of the anisotropic tempered stable process, we obtain the Feynman-Kac representation of the Dirichlet problem with the operator $\mathrmΔ^{α/2,λ}_m$. Therefore, averaging the generated trajectories of the stochastic process leads to the solution of the Dirichlet problem, which is also verified by numerical experiments.

math.PR↗

arXiv2018

Numerical scheme for the Fokker-Planck equations describing anomalous diffusions with two internal states

Recently, the fractional Fokker-Planck equations (FFPEs) with multiple internal states are built for the particles undergoing anomalous diffusion with different waiting time distributions for different internal states, which describe the distribution of positions of the particles [Xu and Deng, Math. Model. Nat. Phenom., $\mathbf{13}$, 10 (2018)]. In this paper, we first develop the Sobolev regularity of the FFPEs with two internal states, including the homogeneous problem with smooth and nonsmooth initial values and the inhomogeneous problem with vanishing initial value, and then we design the numerical scheme for the system of fractional partial differential equations based on the finite element method for the space derivatives and convolution quadrature for the time fractional derivatives. The optimal error estimates of the scheme under the above three different conditions are provided for both space semidiscrete and fully discrete schemes. Finally, one- and two-dimensional numerical experiments are performed to confirm our theoretical analysis and the predicted convergence order.

math.NA↗

arXiv2018

Langevin dynamics for Lévy walk with memory

Memory effects, sometimes, can not be neglected. In the framework of continuous time random walk, memory effect is modeled by the correlated waiting times. In this paper, we derive the two-point probability distribution of the stochastic process with correlated increments as well as the one of its inverse process, and present the Langevin description of Lévy walk with memory, i.e., correlated waiting times. Based on the built Langevin picture, the properties of aging and nonstationary are discussed. The Langevin system exhibits sub-ballistic superdiffusion if the friction force is involved, while it displays super-ballistic diffusion or hyperdiffusion if there is no friction. It is discovered that the correlation of waiting times suppresses the diffusion behavior whether there is friction or not, and the stronger the correlation of waiting times becomes, the slower the diffusion is. In particular, the correlation function, correlation coefficient, ergodicity, and scaling property of the corresponding stochastic process are also investigated.

cond-mat.stat-mech↗

arXiv2018

Central local discontinuous Galerkin method for the space fractional diffusion equation

This paper provides the semi-discrete scheme by the central local discontinuous Galerkin method for space fractional diffusion equation on two sets of overlapping cells, and then we give the stability analysis and error estimates for the scheme. Lastly, we verify the effectiveness of the proposed scheme by the one- and two-dimensional numerical experiments.

math.NA↗

arXiv2018

Renewal theory with fat tailed distributed sojourn times: typical versus rare

Renewal processes with heavy-tailed power law distributed sojourn times are commonly encountered in physical modelling and so typical fluctuations of observables of interest have been investigated in detail. 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 occupation time, and the time interval straddling the observation 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{é}vy and Lamperti laws.

cond-mat.stat-mech↗

arXiv2018

Lévy-walk-like Langevin dynamics

Continuous time random walks and Langevin equations are two classes of stochastic models for describing the dynamics of particles in the natural world. While some of the processes can be conveniently characterized by both of them, more often one model has significant advantages (or has to be used) compared with the other one. In this paper, we consider the weakly damped Langevin system coupled with a new subordinator|$α$-dependent subordinator with $1<α<2$. We pay attention to the diffusion behaviour of the stochastic process described by this coupled Langevin system, and find the super-ballistic diffusion phenomena for the system with an unconfined potential on velocity but sub-ballistic superdiffusion phenomenon with a confined potential, which is like Lévy walk for long times. One can further note that the two-point distribution of inverse subordinator affects mean square displacement of this coupled weakly damped Langevin system in essential.

cond-mat.stat-mech↗

arXiv2018

Phase Transition Dynamics and Stochastic Resonance in Topologically Confined Nematic Liquid Crystals

Topological defects resulted from boundary constraints in confined liquid crystals have attracted extensive research interests. In this paper, we use numerical simulation to study the phase transition dynamics in the context of stochastic resonance in a bistable liquid crystal device containing defects. This device is made of nematic liquid crystals confined in a shallow square well, and is described by the planar Lebwohl-Lasher model. The stochastic phase transition processes of the system in the presence of a weak oscillating potential is simulated using an over-damped Langevin dynamics. Our simulation results reveal that, depending on system size, the phase transition may follow two distinct pathways: in small systems the pre-existing defect structures at the corners hold until the last stage and there is no newly formed defect point in the bulk during the phase transition, In large systems new defect points appear spontaneously in the bulk and eventually merge with the pre-existing defects at the corners. For both transition pathways stochastic resonance can be observed, but show dramatic difference in their responses to the boundary anchoring strength. In small systems we observe a "sticky-boundary" effect for a certain range of anchoring strength in which the phase transition gets stuck and stochastic resonance becomes de-activated. Our work demonstrates the dynamical interplay among defects, noises, and boundary conditions in confined liquid crystals.

cond-mat.soft↗

arXiv2018

Models for characterizing the transition among anomalous diffusions with different diffusion exponents

Based on the theory of continuous time random walks (CTRW), we build the models of characterizing the transitions among anomalous diffusions with different diffusion exponents, often observed in natural world. In the CTRW framework, we take the waiting time probability density function (PDF) 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 process is presented and the positiveness of the PDF of the position of the particles is strictly proved. 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.

cond-mat.stat-mech↗

arXiv2018

Numerical algorithms of the two-dimensional Feynman-Kac equation for reaction and diffusion processes

This paper provides a finite difference discretization for the backward Feynman-Kac equation, governing the distribution of functionals of the path for a particle undergoing both reaction and diffusion [Hou and Deng, J. Phys. A: Math. Theor., {\bf51}, 155001 (2018)]. Numerically solving the equation with the time tempered fractional substantial derivative and tempered fractional Laplacian consists in discretizing these two non-local operators. Here, using convolution quadrature, we provide a first-order and second-order schemes for discretizing the time tempered fractional substantial derivative, which doesn't require the assumption of the regularity of the solution in time; we use the finite difference method to approximate the two-dimensional tempered fractional Laplacian, and the accuracy of the scheme depends on the regularity of the solution on $\barΩ$ rather than the whole space. Lastly, we verify the predicted convergence orders and the effectiveness of the presented schemes by numerical examples.

math.NA↗