Fluid limit of the continuous-time random walk with general Lévy jump distribution functions

Abstract

The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions ψ(t) and general jump distribution functions η(x). There are two well-known fluid limits of this model in the uncoupled case. For exponential decaying waiting times and Gaussian jump distribution functions the fluid limit leads to the diffusion equation. On the other hand, for algebraic decaying waiting times, ψ ∼ t −(1+β) , and algebraic decaying jump distributions, η ∼ x −(1+α) , corresponding to Lévy stable processes, the fluid limit leads to the fractional diffusion equation of order α in space and order β in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Lévy stochastic processes in the Lévy-Khintchine representation for the jump distribution function and obtain an integro-differential equation describing the dynamics in the fluid limit. The resulting equation contains as special cases the regular and the fractional diffusion equations. As an application we consider the case of CTRWs with exponentially truncated Lévy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describes the interplay between memory, long jumps, and truncation effects in the intermediate asymptotic regime. The dynamics exhibits a transition from superdiffusion to subdiffusion with the crossover time scaling as τ c ∼ λ −α/β where 1/λ is the truncation length scale. The asymptotic behavior of the propagator (Green's function) of the truncated fractional equation exhibits a transition from algebraic decay for t τ c to stretched Gaussian decay for t τc.

Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation

“…2, 3, 4, 5, 6, 7, 8, 9, 10, applications and enhancements of these techniques were presented. The relevance of fractional calculus in the phenomenological description of anomalous diffusion has been discussed within applications of statistical mechanics in physics, chemistry and biology [11,12,13,14,15,16,17] as well as finance [18,19,20,21,22]; even human travel and the spreading of epidemics were modeled with fractional diffusion [23]. A direct Monte Carlo approach to fractional Fokker-Planck dynamics through the underlying CTRW requires random numbers drawn from the Mittag-Leffler distribution.…”

Particle-tracking simulation of fractional diffusion-reaction processes

“…The space fractional-order PDE [7,17,18] can efficiently capture super-diffusive transport of nonreactive tracers observed in complex, natural geological formations, as reviewed and demonstrated further by Zhang et al [12]. The space tempered stable model [19,20,21] is the logic extension of the standard fractional-order PDE to capture the convergent spatial moments and the natural cutoff of power-law distributions present in real physical systems. The model describes the transition from super-diffusion to asymptotic, normal diffusion limits over time.…”

“…Note that when dτ j = 0 in the time-Langevin equation (20), the above Lagrangian method simulates the forward reaction with transport governed by the classical time-fractional model (19). A zero operational time dτ j implies instantaneous jumps [10], or mobile status all the time [32].…”

“…The standard fractional-order PDE and the classical diffusion equation are two end-members of the space tempered stable model. The following fractional-order, advection-dispersion-reaction (ADR) equation may describe the elementary, bimolecular secondorder reversible reaction A + B ⇋ C undergoing tempered superdiffusive transport: ∂x α, λ denotes the tempered fractional derivative [19,20]. For example:…”

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

“…(44) changes its properties at |x| = L and one can expect a different diffusion behaviour in the surface region, compared to the bulk. If L is sufficiently large, the dynamics resolves itself to the truncated Lévy flights and the diffusion properties are equivalent to the Gaussian case [46], providing the time is relatively small; then one gets the standard anomalous diffusion law,…”