Anomalous diffusion with linear reaction dynamics: from continuous time random walks to fractional reaction-diffusion equations.

We have revisited the problem of anomalously diffusing species, modeled at the mesoscopic level using continuous time random walks, to include linear reaction dynamics. If a constant proportion of walkers are added or removed instantaneously at the start of each step then the long time asymptotic limit yields a fractional reaction-diffusion equation with a fractional order temporal derivative operating on both the standard diffusion term and a linear reaction kinetics term. If the walkers are added or removed at a constant per capita rate during the waiting time between steps then the long time asymptotic limit has a standard linear reaction kinetics term but a fractional order temporal derivative operating on a nonstandard diffusion term. Results from the above two models are compared with a phenomenological model with standard linear reaction kinetics and a fractional order temporal derivative operating on a standard diffusion term. We have also developed further extensions of the CTRW model to include more general reaction dynamics.

[1]  W. Coffey,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2002 .

[2]  Melvin Lax,et al.  Stochastic Transport in a Disordered Solid. I. Theory , 1973 .

[3]  Brian Berkowitz,et al.  Morphogen gradient formation in a complex environment: an anomalous diffusion model. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Enrico Scalas,et al.  Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[6]  M. Shlesinger,et al.  Stochastic pathway to anomalous diffusion. , 1987, Physical review. A, General physics.

[7]  KARL PEARSON,et al.  The Problem of the Random Walk , 1905, Nature.

[8]  Brian Berkowitz,et al.  Continuous time random walks revisited: first passage time and spatial distributions , 2004 .

[9]  Kazuhiko Seki,et al.  Fractional reaction-diffusion equation , 2003 .

[10]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[11]  J. Bouchaud,et al.  Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications , 1990 .

[12]  I. Podlubny.  Fractional differential equations , 1998 .

[13]  J. Klafter,et al.  The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , 2004 .

[14]  S. Havlin,et al.  Diffusion and Reactions in Fractals and Disordered Systems , 2000 .

[15]  Sangyoub Lee,et al.  Fractional dynamics approach to diffusion-assisted reactions in disordered media , 2002 .

[16]  Subdiffusion and the cage effect studied near the colloidal glass transition , 2001, cond-mat/0111073.

[17]  Daniel Campos,et al.  Front propagation in reaction-dispersal models with finite jump speed. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  S. Wearne,et al.  Fractional Reaction-Diffusion , 2000 .

[19]  V. Méndez,et al.  Front propagation in hyperbolic fractional reaction-diffusion equations. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  S L Wearne,et al.  Turing pattern formation in fractional activator-inhibitor systems. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  Akihiro Kusumi,et al.  Detection of non-Brownian diffusion in the cell membrane in single molecule tracking. , 2005, Biophysical journal.

[22]  R. Gorenflo,et al.  Fractional calculus and continuous-time finance II: the waiting-time distribution , 2000, cond-mat/0006454.

[23]  Hari M. Srivastava,et al.  The H-functions of one and two variables, with applications , 1982 .

[24]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

[25]  I M Sokolov,et al.  From diffusion to anomalous diffusion: a century after Einstein's Brownian motion. , 2005, Chaos.

[26]  Vicenç Méndez,et al.  Continuous-time random walks and traveling fronts. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  David A. Benson,et al.  Advection and dispersion in time and space , 2002 .

[28]  A. Compte,et al.  The generalized Cattaneo equation for the description of anomalous transport processes , 1997 .

[29]  Francesco Mainardi,et al.  The fractional Fick's law for non-local transport processes , 2001 .

[30]  J. Ross,et al.  Systematic derivation of reaction-diffusion equations with distributed delays and relations to fractional reaction-diffusion equations and hyperbolic transport equations: application to the theory of Neolithic transition. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  S. Wearne,et al.  Existence of Turing Instabilities in a Two-Species Fractional Reaction-Diffusion System , 2002, SIAM J. Appl. Math..

[32]  Katja Lindenberg,et al.  Reaction front in an A+B-->C reaction-subdiffusion process. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[34]  Tommy Nilsson,et al.  Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy. , 2003, Biophysical journal.

[35]  T. A. M. Langlands,et al.  Solution of a modified fractional diffusion equation , 2006 .

[36]  I M Sokolov,et al.  Reaction-subdiffusion equations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[37]  K. Seki,et al.  Recombination kinetics in subdiffusive media , 2003 .

[38]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[39]  M. Saxton,et al.  Anomalous subdiffusion in fluorescence photobleaching recovery: a Monte Carlo study. , 2001, Biophysical journal.

[40]  D. Zanette,et al.  Experimental evidence of power-law trapping-time distributions in porous media. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[41]  E. Montroll,et al.  Random Walks on Lattices. II , 1965 .