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Title: Fluid limit of the continuous-time 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 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 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 Levy-Khintchine 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 » 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).« less

Authors:
 [1];  [2]
  1. Birbeck College, University of London
  2. 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:  
DE-AC05-00OR22725
Resource Type:
Journal Article
Journal Name:
Physical Review E
Additional Journal Information:
Journal Volume: 76; Journal Issue: 4; Journal ID: ISSN 1539-3755
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; STOCHASTIC-PROCESS; ULTRASLOW CONVERGENCE; FRACTIONAL DIFFUSION; PLASMA TURBULENCE; FLIGHT; EQUATIONS; DYNAMICS; MODELS

Citation Formats

Cartea, A., and Del-Castillo-Negrete, Diego B. Fluid limit of the continuous-time random walk with general Levy jump distribution functions. United States: N. p., 2007. Web. doi:10.1103/PhysRevE.76.041105.
Cartea, A., & Del-Castillo-Negrete, Diego B. Fluid limit of the continuous-time random walk with general Levy jump distribution functions. United States. doi:10.1103/PhysRevE.76.041105.
Cartea, A., and Del-Castillo-Negrete, Diego B. Mon . "Fluid limit of the continuous-time random walk with general Levy jump distribution functions". United States. doi:10.1103/PhysRevE.76.041105.
@article{osti_1032022,
title = {Fluid limit of the continuous-time random walk with general Levy jump distribution functions},
author = {Cartea, A. and Del-Castillo-Negrete, 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 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 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 Levy-Khintchine 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 = {1539-3755},
number = 4,
volume = 76,
place = {United States},
year = {2007},
month = {1}
}