Heterogeneous continuous-time random walks.

We introduce a heterogeneous continuous-time random walk (HCTRW) model as a versatile analytical formalism for studying and modeling diffusion processes in heterogeneous structures, such as porous or disordered media, multiscale or crowded environments, weighted graphs or networks. We derive the exact form of the propagator and investigate the effects of spatiotemporal heterogeneities onto the diffusive dynamics via the spectral properties of the generalized transition matrix. In particular, we show how the distribution of first-passage times changes due to local and global heterogeneities of the medium. The HCTRW formalism offers a unified mathematical language to address various diffusion-reaction problems, with numerous applications in material sciences, physics, chemistry, biology, and social sciences.

[1]  Brian Berkowitz,et al.  Anomalous transport in laboratory‐scale, heterogeneous porous media , 2000 .

[2]  Bin Wu,et al.  Eigenvalues of normalized Laplacian matrices of fractal trees and dendrimers: analytical results and applications. , 2013, The Journal of chemical physics.

[3]  Katja Lindenberg,et al.  Reaction-subdiffusion and reaction-superdiffusion equations for evanescent particles performing continuous-time random walks. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Seungoh Ryu,et al.  Determining multiple length scales in rocks , 2000, Nature.

[5]  Andrey G. Cherstvy,et al.  Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. , 2014, Physical chemistry chemical physics : PCCP.

[6]  Minh Tan Vu,et al.  Reactive transport in porous media: pore-network model approach compared to pore-scale model. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[8]  A. Zoia,et al.  Nonlinear random-walk approach to concentration-dependent contaminant transport in porous media. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  J. Kirchner,et al.  Fractal stream chemistry and its implications for contaminant transport in catchments , 2000, Nature.

[10]  Sergio Gómez,et al.  Spectral properties of the Laplacian of multiplex networks , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  M. Sternheim,et al.  Non-Hermitian Hamiltonians, Decaying States, and Perturbation Theory , 1972 .

[12]  Competing reaction processes on a lattice as a paradigm for catalyst deactivation. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  C. Broeck,et al.  Decay times in one-dimensional chains , 1986 .

[14]  Ralf Metzler,et al.  Physical pictures of transport in heterogeneous media: Advection‐dispersion, random‐walk, and fractional derivative formulations , 2002, cond-mat/0202327.

[15]  D. Grebenkov,et al.  The escape problem for mortal walkers. , 2016, The Journal of chemical physics.

[16]  Melvin Lax,et al.  Stochastic Transport in a Disordered Solid. II. Impurity Conduction , 1973 .

[17]  Dubois,et al.  Pore network as a model of porous media: comparison between nonhierarchical and hierarchical organizations of pores , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[18]  M. Stampanoni,et al.  Topology of evolving pore networks , 2012 .

[19]  A. Roberts,et al.  Continuum diffusion on networks: trees with hyperbranched trunks and fractal branches. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[20]  Youyi Jiang,et al.  Estimation of the eigenvalues and the smallest singular value of matrices , 2010 .

[21]  E. Barkai.  CTRW pathways to the fractional diffusion equation , 2001, cond-mat/0108024.

[22]  Shuigeng Zhou,et al.  Exact solution for mean first-passage time on a pseudofractal scale-free web. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Andrey G. Cherstvy,et al.  Anomalous, non-Gaussian tracer diffusion in crowded two-dimensional environments , 2015, 1508.02029.

[24]  Cecilia Mascolo,et al.  Mining open datasets for transparency in taxi transport in metropolitan environments , 2015, EPJ Data Science.

[25]  Igor M. Sokolov,et al.  Field-induced dispersion in subdiffusion. , 2006 .

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

[27]  Elliott W. Montroll,et al.  Random walks on lattices. IV. Continuous-time walks and influence of absorbing boundaries , 1973 .

[28]  Peter Grassberger,et al.  Localization transition of biased random walks on random networks. , 2007, Physical review letters.

[29]  Survival probability of an immobile target in a sea of evanescent diffusive or subdiffusive traps: a fractional equation approach. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  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 .

[31]  J. J. Kozak,et al.  Lattice statistical theory of random walks on a fractal-like geometry. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[32]  S. Redner,et al.  Mortality, redundancy, and diversity in stochastic search. , 2015, Physical review letters.

[33]  D. Grebenkov,et al.  The role of morphology in mathematical models of placental gas exchange. , 2016, Journal of applied physiology.

[34]  B Kahng,et al.  First passage time for random walks in heterogeneous networks. , 2012, Physical review letters.

[35]  H. Wio,et al.  Narrow-escape-time problem: the imperfect trapping case. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[36]  K. Mecke,et al.  Splitting of the Universality Class of Anomalous Transport in Crowded Media. , 2016, Physical review letters.

[37]  L. Dagdug,et al.  Diffusion coefficients for two-dimensional narrow asymmetric channels embedded on flat and curved surfaces , 2014 .

[38]  N. Lequeux,et al.  Microstructure evolution of hydrated cement pastes. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Albert-László Barabási,et al.  Statistical mechanics of complex networks , 2001, ArXiv.

[40]  L. Trefethen,et al.  Spectra and Pseudospectra , 2020 .

[41]  S. B. Yuste,et al.  Exploration and trapping of mortal random walkers. , 2013, Physical review letters.

[42]  M. Dentz,et al.  Transport behavior of coupled continuous-time random walks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Marc Barthelemy,et al.  Spatial Networks , 2010, Encyclopedia of Social Network Analysis and Mining.

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

[45]  Solvable continuous-time random walk model of the motion of tracer particles through porous media. , 2016, Physical review. E.

[46]  Delio Mugnolo.  Semigroup Methods for Evolution Equations on Networks , 2014 .

[47]  Denis S. Grebenkov,et al.  Partially Reflected Brownian Motion: A Stochastic Approach to Transport Phenomena , 2006, math/0610080.

[48]  V. M. Kenkre,et al.  Generalized master equations for continuous-time random walks , 1973 .

[49]  Reuven Cohen,et al.  Anomalous biased diffusion in networks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  Joseph W. Haus,et al.  Diffusion in regular and disordered lattices , 1987 .

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

[52]  Alessandro Vespignani,et al.  Random walks and search in time-varying networks. , 2012, Physical review letters.

[53]  Jari Saramäki,et al.  Temporal Networks , 2011, Encyclopedia of Social Network Analysis and Mining.

[54]  Ofer Zeitouni,et al.  Random Walks in Random Environment , 2009, Encyclopedia of Complexity and Systems Science.

[55]  M. Sahimi.  Flow phenomena in rocks : from continuum models to fractals, percolation, cellular automata, and simulated annealing , 1993 .

[56]  Guillemette Picard,et al.  Method for modeling transport of particles in realistic porous networks: application to the computation of NMR flow propagators. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[57]  Elena Agliari,et al.  Two-particle problem in comblike structures. , 2016, Physical review. E.

[58]  J. Goutsias,et al.  Markovian dynamics on complex reaction networks , 2012, 1205.5524.

[59]  A. Talbot.  The Accurate Numerical Inversion of Laplace Transforms , 1979 .

[60]  Brian Berkowitz,et al.  Anomalous transport in correlated velocity fields. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[61]  S. Havlin,et al.  Scaling theory of transport in complex biological networks , 2007, Proceedings of the National Academy of Sciences.

[62]  J. Masoliver,et al.  Continuous time persistent random walk: a review and some generalizations , 2017 .

[63]  Denis S Grebenkov,et al.  Subdiffusion in a bounded domain with a partially absorbing-reflecting boundary. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[64]  M. Tachiya,et al.  Partially diffusion‐controlled recombination , 1979 .

[65]  Kamel Hooman,et al.  Evaluating transport in irregular pore networks. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[66]  S. B. Yuste,et al.  Evanescent continuous-time random walks. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[67]  Alessandro Vespignani,et al.  Reaction–diffusion processes and metapopulation models in heterogeneous networks , 2007, cond-mat/0703129.

[68]  Heiko Rieger,et al.  Random walks on complex networks. , 2004, Physical review letters.

[69]  P. Bressloff,et al.  Stochastic models of intracellular transport , 2013 .

[70]  Elliott W. Montroll,et al.  Random Walks on Lattices. III. Calculation of First‐Passage Times with Application to Exciton Trapping on Photosynthetic Units , 1969 .

[71]  M. Sahimi.  Dispersion in porous media, continuous-time random walks, and percolation. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[72]  Ruben Juanes,et al.  Anomalous physical transport in complex networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[73]  M. Bálek,et al.  Large Networks and Graph Limits , 2022 .

[74]  M V Tamm,et al.  Anomalous diffusion in fractal globules. , 2014, Physical review letters.

[75]  P. Levitz.  Statistical Modeling of Pore Networks , 2008 .

[76]  Marc Timme,et al.  Small-world spectra in mean field theory , 2012, Physical review letters.

[77]  Ruben Juanes,et al.  Spatial Markov model of anomalous transport through random lattice networks. , 2011, Physical review letters.

[78]  J. Hunter.  Stationary distributions of perturbed Markov chains , 1986 .

[79]  Shun Wang,et al.  Stick-slip behaviours of dry glass beads in triaxial compression , 2017 .

[80]  A. Vespignani,et al.  The architecture of complex weighted networks. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[81]  Vito Latora,et al.  Characteristic times of biased random walks on complex networks , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[83]  E. Villaseñor.  Introduction to Quantum Mechanics , 2008, Nature.

[84]  The number statistics and optimal history of non-equilibrium steady states of mortal diffusing particles , 2015, 1502.02813.

[85]  Andrey G. Cherstvy,et al.  Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes , 2013, 1303.5533.

[86]  Carsten Carstensen,et al.  Inclusion of the roots of a polynomial based on Gerschgorin's theorem , 1991 .

[87]  M. Coppens.  The effect of fractal surface roughness on diffusion and reaction in porous catalysts - From fundamentals to practical applications , 1999 .

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

[89]  C. Angstmann,et al.  Continuous-time random walks on networks with vertex- and time-dependent forcing. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[90]  Zhongzhi Zhang,et al.  Random walks in weighted networks with a perfect trap: an application of Laplacian spectra. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[91]  J. Stoyanov.  A Guide to First‐passage Processes , 2003 .

[92]  J. Klafter,et al.  The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times , 2001, cond-mat/0202326.

[93]  H. Scher,et al.  Transport in disordered media with spatially nonuniform fields. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

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