Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary.

The exit time of a subdiffusive process from a bounded domain with a partially absorbing/reflecting boundary is considered. The short-time and long-time behaviors of the exit time probability density are investigated by using a spectral decomposition on the basis of the Laplace operator eigenfunctions. Rotation-invariant domains are analyzed in depth in order to illustrate the use of theoretical formulas and to compare them to numerical simulations. The asymptotic results obtained are relevant for describing subdiffusion inside a living cell with a semipermeable membrane, in a chemical reactor filled with catalytic grains of finite reactivity, or in mineral or biological samples which are probed by nuclear magnetic resonance measurements subject to surface relaxation.

[1]  K. M. Kolwankar,et al.  Brownian flights over a fractal nest and first-passage statistics on irregular surfaces. , 2006, Physical review letters.

[2]  F. Mainardi,et al.  Fox H functions in fractional diffusion , 2005 .

[3]  Denis S. Grebenkov,et al.  NMR survey of reflected brownian motion , 2007 .

[4]  S. Bochner.  Diffusion Equation and Stochastic Processes. , 1949, Proceedings of the National Academy of Sciences of the United States of America.

[5]  G. Weiss,et al.  Transport properties of the continuous-time random walk with a long-tailed waiting-time density , 1989 .

[6]  From the solutions of diffusion equation to the solutions of subdiffusive one , 2004, cond-mat/0404342.

[7]  Ralf Metzler,et al.  Subdiffusion and weak ergodicity breaking in the presence of a reactive boundary. , 2007, Physical review letters.

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

[9]  A. Szabó,et al.  Role of diffusion in ligand binding to macromolecules and cell-bound receptors. , 1982, Biophysical journal.

[10]  Joseph Klafter,et al.  Probability Distributions for Continuous-Time Random Walks with Long Tails , 1994 .

[11]  Francesco Mainardi,et al.  On Mittag-Leffler-type functions in fractional evolution processes , 2000 .

[12]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[13]  Klaus Schulten,et al.  First passage time approach to diffusion controlled reactions , 1980 .

[14]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies) , 2006 .

[15]  M. Weiss,et al.  Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. , 2004, Biophysical journal.

[16]  Rainer Kimmich,et al.  Strange kinetics, porous media, and NMR , 2002 .

[17]  H. Risken.  Fokker-Planck Equation , 1996 .

[18]  E. Cox,et al.  Physical nature of bacterial cytoplasm. , 2006, Physical review letters.

[19]  S. Havlin,et al.  Diffusion in disordered media , 2002 .

[20]  B. Sapoval,et al.  Diffusional screening in real 3D human acini—a theoretical study , 2005, Respiratory Physiology & Neurobiology.

[21]  Multifractal properties of the harmonic measure on Koch boundaries in two and three dimensions. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Y. Chiew,et al.  Computer simulation of diffusion‐controlled reactions in dispersions of spherical sinks , 1989 .

[23]  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 .

[24]  Mark M. Meerschaert,et al.  Limit theorems for continuous-time random walks with infinite mean waiting times , 2004, Journal of Applied Probability.

[25]  H. Risken.  The Fokker-Planck equation : methods of solution and applications , 1985 .

[26]  G. Weiss.  Overview of theoretical models for reaction rates , 1986 .

[27]  G. Zaslavsky.  Chaos, fractional kinetics, and anomalous transport , 2002 .

[28]  O Bénichou,et al.  First-passage time distributions for subdiffusion in confined geometry. , 2007, Physical review letters.

[29]  B. Sapoval,et al.  Can one hear the shape of an electrode? II. Theoretical study of the Laplacian transfer , 1999 .

[30]  Katja Lindenberg,et al.  Subdiffusive target problem: survival probability. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  D. Grebenkov.  What makes a boundary less accessible. , 2005, Physical review letters.

[32]  O Bénichou,et al.  First-exit times and residence times for discrete random walks on finite lattices. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  Z. Schuss,et al.  0 41 20 89 v 1 2 5 D ec 2 00 4 Stochastic Chemical Reactions in Micro-domains , 2008 .

[34]  S. Redner.  A guide to first-passage processes , 2001 .

[35]  Jaeyoung Sung,et al.  Exact dynamics of a continuous time random walker in the presence of a boundary: beyond the intuitive boundary condition approach. , 2003, Physical review letters.

[36]  M. Fixman,et al.  General theory of diffusion‐controlled reactions , 1973 .

[37]  D. Grebenkov.  SCALING PROPERTIES OF THE SPREAD HARMONIC MEASURES , 2006 .

[38]  How to measure subdiffusion parameters. , 2005, Physical review letters.

[39]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1951 .

[40]  D. Grebenkov.  Searching for partially reactive sites: Analytical results for spherical targets. , 2010, The Journal of chemical physics.

[41]  Barkai,et al.  From continuous time random walks to the fractional fokker-planck equation , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[42]  J. Klafter,et al.  First-passage times in complex scale-invariant media , 2007, Nature.

[43]  D. Grebenkov.  Residence times and other functionals of reflected Brownian motion. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[44]  A. Grosberg,et al.  First passage times and asymmetry of DNA translocation. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Sidney Redner,et al.  A guide to first-passage processes , 2001 .

[46]  J. Klafter,et al.  Probing microscopic origins of confined subdiffusion by first-passage observables , 2008, Proceedings of the National Academy of Sciences.

[47]  K. Brownstein,et al.  Importance of classical diffusion in NMR studies of water in biological cells , 1979 .

[48]  D. Reichman,et al.  The subdiffusive targeting problem. , 2008, The journal of physical chemistry. B.

[49]  George E. Kimball,et al.  Diffusion-controlled reaction rates , 1949 .

[50]  Ralf Metzler,et al.  Boundary value problems for fractional diffusion equations , 2000 .

[51]  E. Barkai,et al.  Fractional Fokker-Planck equation, solution, and application. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[52]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[53]  I. Sokolov.  Lévy flights from a continuous-time process. , 2000, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[55]  M. E. Muller.  Some Continuous Monte Carlo Methods for the Dirichlet Problem , 1956 .

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

[57]  O Bénichou,et al.  First-passage times for random walks in bounded domains. , 2005, Physical review letters.

[58]  Spectral properties of the Brownian self-transport operator , 2002 .

[59]  Lee,et al.  Random-walk simulation of diffusion-controlled processes among static traps. , 1989, Physical review. B, Condensed matter.

[60]  Amit Singer,et al.  Partially Reflected Diffusion , 2008, SIAM J. Appl. Math..

[61]  B. Sapoval,et al.  Mathematical basis for a general theory of Laplacian transport towards irregular interfaces. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[62]  Sapoval.  General formulation of Laplacian transfer across irregular surfaces. , 1994, Physical review letters.

[63]  J. Michael Schurr.  The role of diffusion in enzyme kinetics. , 1970, Biophysical journal.

[64]  Tom Chou,et al.  Multistage adsorption of diffusing macromolecules and viruses. , 2007, The Journal of chemical physics.

[65]  David Holcman,et al.  Survival probability of diffusion with trapping in cellular neurobiology. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.