Generalized Continuous Time Random Walks, Master Equations, and Fractional Fokker--Planck Equations

Abstract

Abstract. Continuous time random walks, which generalize random walks by adding a stochastic time between jumps, provide a useful description of stochastic transport at mesoscopic scales. The continuous time random walk model can accommodate certain features, such as trapping, which are not manifest in the standard macroscopic diffusion equation. The trapping is incorporated through a waiting time density, and a fractional diffusion equation results from a power law waiting time. A generalized continuous time random walk model with biased jumps has been used to consider transport that is also subject to an external force. Here we have derived the master equations for continuous time random walks with space-and time-dependent forcing for two cases: when the force is evaluated at the start of the waiting time and at the end of the waiting time. The differences persist in low order spatial continuum approximations; however, the two processes are shown to be governed by the same Fokker-Planck equations in the diffusion limit. Thus the fractional FokkerPlanck equation with space-and time-dependent forcing is robust to these changes in the underlying stochastic process. 1. Introduction. A continuous time random walk (CTRW) is a stochastic process where a particle arrives at a position, waits for a stochastic time, and stochastically jumps to a new position. The motion of the particle is governed by waiting time, and step length, probability densities [40]. Generalized CTRWs have been used as a stochastic basis for deriving fractional diffusion equations [16,22,36], fractional Cattenao equations [12], fractional reaction diffusion equations [53,20,39,14,52,3,49], fractional cable equations [21,27,28], fractional Fokker-Planck equations (FFPEs) [35,46,8,3], and fractional chemotaxis equations [26,15]. The evolution equation for the probability density function (PDF) of the random walking particles in a CTRW can be written as generalized master equations (GMEs) [45,24]. If the waiting times are exponentially distributed and the step length density is Gaussian, then in the diffusion limit the evolution equation for the PDF of the random walking particles is the standard diffusion equation. The CTRW also allows for anomalous diffusion in which the mean squared displacement increases slower (subdiffusion) or faster (superdiffusion) than linearly with time. The canonical case is a fractional power of time x 2 (t) ∼ t γ where γ = 1 [36]. Subdiffusion (0 < γ < 1) arises from CTRWs with power law waiting time densities such as Pareto or Mittag-Leffler waiting time densities. In this case, in the diffusion limit the evolution equation for the particle

Numerical Solution of the Time-Fractional Fokker--Planck Equation with General Forcing

“…The experiments give no evidence that the methods fail if 0 < α ≤ 1/2, although the convergence rate deteriorates as α decreases when using a uniform time step. We also apply our method to a problem from a recent paper by Angstmann et al [1] and investigate whether the regularity of the initial data affects the stability of the methods. A brief appendix proves a technical result (Lemma A.2) used in showing stability of the time-stepping procedure.…”

“…Although we do not know an analytical solution, Laplace transform techniques [1] show that in the limiting case when L → ∞, and interpreting u(·, t) as a probability density function, the expected position, or first moment, is x(t) = To investigate whether the stability properties really depend on the smoothness of the initial data, we solved the same problem as in Section 5.2 but chose the nodal values for the discrete initial data u 0h to be uniformly distributed pseudorandom numbers in the unit interval. For 0 ≤ t ≤ T = 40 and many different combinations of N and P , we never observed any kind of instability.…”

“…For the initial data u 0 , we chose a normal probability density function with mean 0 and variance σ 2 . This choice of F is taken from a recent paper by Angstmann et al [1]; notice that F x = −1 < 0 so the first assumption of Theorem 3.2 is not satisfied and we must rely on Theorem 3.3 to ensure stability of the spatially discrete scheme (3.2). For our computations, we used the values α = 0.75 and σ = 0.5, with a mesh grading parameter γ = 1/α.…”

A Semidiscrete Finite Element Approximation of a Time-Fractional Fokker--Planck Equation with NonSmooth Initial Data

“…We consider the spatial discretisation via Galerkin finite elements of a time-fractional Fokker-Planck equation [1,13], ∂ t u − ∇ · ∂ 1−α t κ α ∇u − F∂ 1−α t u = 0 for x ∈ Ω and 0 < t < T ,…”

Stationary Equations for Non-Markovian Biochemical Systems

“…Non-Markovity has also been verified by the increasing availability of time-resolved data on different kinds of interactions [23][24][25][26][27][28][29]. The continuous time random walk (CRTW) provides a systematic starting point to account for arbitrary waiting time distributions between reaction events [30][31][32][33][34][35][36][37][38].…”