Fluid limit of the continuoustime random walk with general Levy 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 psi(t) and general jump distribution functions eta(x). There are two wellknown 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 psi similar to t((1+beta)) and algebraic decaying jump distributions eta similar to x((1+alpha)) corresponding to Levy stable processes, the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Levy stochastic processes in the LevyKhintchine representation for the jump distribution function and obtain an integrodifferential 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 Levy jump distribution functions. In this case the fluid limit leads to a transport equation with exponentially truncated fractional derivatives which describesmore »
 Authors:

 Birbeck College, University of London
 ORNL
 Publication Date:
 Research Org.:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC)
 OSTI Identifier:
 1032022
 DOE Contract Number:
 DEAC0500OR22725
 Resource Type:
 Journal Article
 Journal Name:
 Physical Review E
 Additional Journal Information:
 Journal Volume: 76; Journal Issue: 4; Journal ID: ISSN 15393755
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; DECAY; DIFFUSION EQUATIONS; DISTRIBUTION FUNCTIONS; PROPAGATOR; STOCHASTIC PROCESSES; TRANSPORT; ANOMALOUS DIFFUSION; STOCHASTICPROCESS; ULTRASLOW CONVERGENCE; FRACTIONAL DIFFUSION; PLASMA TURBULENCE; FLIGHT; EQUATIONS; DYNAMICS; MODELS
Citation Formats
Cartea, A., and DelCastilloNegrete, Diego B. Fluid limit of the continuoustime random walk with general Levy jump distribution functions. United States: N. p., 2007.
Web. doi:10.1103/PhysRevE.76.041105.
Cartea, A., & DelCastilloNegrete, Diego B. Fluid limit of the continuoustime random walk with general Levy jump distribution functions. United States. doi:10.1103/PhysRevE.76.041105.
Cartea, A., and DelCastilloNegrete, Diego B. Mon .
"Fluid limit of the continuoustime random walk with general Levy jump distribution functions". United States. doi:10.1103/PhysRevE.76.041105.
@article{osti_1032022,
title = {Fluid limit of the continuoustime random walk with general Levy jump distribution functions},
author = {Cartea, A. and DelCastilloNegrete, Diego B},
abstractNote = {The continuous time random walk (CTRW) is a natural generalization of the Brownian random walk that allows the incorporation of waiting time distributions psi(t) and general jump distribution functions eta(x). There are two wellknown 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 psi similar to t((1+beta)) and algebraic decaying jump distributions eta similar to x((1+alpha)) corresponding to Levy stable processes, the fluid limit leads to the fractional diffusion equation of order alpha in space and order beta in time. However, these are two special cases of a wider class of models. Here we consider the CTRW for the most general Levy stochastic processes in the LevyKhintchine representation for the jump distribution function and obtain an integrodifferential 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 Levy 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 tau(c)similar to lambda(alpha/beta), where 1/lambda 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 <>tau(c).},
doi = {10.1103/PhysRevE.76.041105},
journal = {Physical Review E},
issn = {15393755},
number = 4,
volume = 76,
place = {United States},
year = {2007},
month = {1}
}