Anomalous diffusion in correlated continuous time random walks

We demonstrate that continuous time random walks in which successive waiting times are correlated by Gaussian statistics lead to anomalous diffusion with the mean squared displacement r2(t) t2/3. Long-ranged correlations of the waiting times with a power-law exponent ? (0 < ? ? 2) give rise to subdiffusion of the form r2(t) t?/(1 + ?). In contrast, correlations in the jump lengths are shown to produce superdiffusion. We show that in both cases weak ergodicity breaking occurs. Our results are in excellent agreement with simulations.

[1]  E. Montroll,et al.  Anomalous transit-time dispersion in amorphous solids , 1975 .

[2]  Tanguy Le Borgne,et al.  Lagrangian statistical model for transport in highly heterogeneous velocity fields. , 2008, Physical review letters.

[3]  H. Larralde,et al.  Lévy walk patterns in the foraging movements of spider monkeys (Ateles geoffroyi) , 2003, Behavioral Ecology and Sociobiology.

[4]  T. Geisel,et al.  The scaling laws of human travel , 2006, Nature.

[5]  Ralf Metzler,et al.  Lévy strategies in intermittent search processes are advantageous , 2008, Proceedings of the National Academy of Sciences.

[6]  J. Klafter,et al.  Nonergodicity mimics inhomogeneity in single particle tracking. , 2008, Physical review letters.

[7]  Kharkov,et al.  Fractional kinetics for relaxation and superdiffusion in a magnetic field , 2002 .

[8]  F. Lillo,et al.  The non-random walk of stock prices: the long-term correlation between signs and sizes , 2007, 0711.4596.

[9]  F Bartumeus,et al.  Optimizing the encounter rate in biological interactions: Lévy versus Brownian strategies. , 2002, Physical review letters.

[10]  Igor M. Sokolov,et al.  ANOMALOUS TRANSPORT IN EXTERNAL FIELDS : CONTINUOUS TIME RANDOM WALKS AND FRACTIONAL DIFFUSION EQUATIONS EXTENDED , 1998 .

[11]  Mark M. Meerschaert,et al.  Correlated continuous time random walks , 2008, 0809.1612.

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

[13]  R. Metzler,et al.  Random time-scale invariant diffusion and transport coefficients. , 2008, Physical review letters.

[14]  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.

[15]  E. Barkai,et al.  Distribution of time-averaged observables for weak ergodicity breaking. , 2007, Physical review letters.

[16]  Miquel Montero,et al.  Nonindependent continuous-time random walks. , 2007, 0707.4087.

[17]  R. Barkana.  On correlated random walks and 21-cm fluctuations during cosmic reionization , 2007, 0704.3534.

[18]  Solomon,et al.  Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. , 1993, Physical review letters.

[19]  Golan Bel,et al.  Weak Ergodicity Breaking in the Continuous-Time Random Walk , 2005 .

[20]  Ludovico Silvestri,et al.  Event-driven power-law relaxation in weak turbulence. , 2009, Physical review letters.

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

[22]  C. Peng,et al.  Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics. , 1996, The American journal of physiology.

[23]  Jenny Nelson,et al.  Continuous-time random-walk model of electron transport in nanocrystalline TiO 2 electrodes , 1999 .

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

[25]  Ralf Metzler,et al.  Optimal target search on a fast-folding polymer chain with volume exchange. , 2005, Physical review letters.

[26]  J. Klafter,et al.  Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach , 1999 .

[27]  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 .

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

[29]  Igor M Sokolov,et al.  Continuous-time random walk with correlated waiting times. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.