Inverse cascade in a percolation model: hierarchical description of time-dependent scaling.

The dynamics of a two-dimensional site percolation model on a square lattice is studied using the hierarchical approach introduced by Gabrielov [Phys. Rev. E 60, 5293 (1999)]. The key elements of the approach are the tree representation of clusters and their coalescence, and the Horton-Strahler scheme for cluster ranking. Accordingly, the evolution of the percolation model is considered as a hierarchical inverse cascade of cluster aggregation. A three-exponent time-dependent scaling for the cluster rank distribution is derived using the Tokunaga branching constraint and classical results on percolation in terms of cluster masses. Deviations from the pure scaling are described. An empirical constraint on the dynamics of a rank population is reported based on numerical simulations.

[1]  S. Redner,et al.  Introduction To Percolation Theory , 2018 .

[2]  Carsten Riedel,et al.  International Handbook of Earthquake and Engineering Seismology , 2006 .

[3]  William I. Newman,et al.  An inverse cascade model for self-organized complexity and natural hazards , 2005 .

[4]  刘金明,et al.  IL-13受体α2降低血吸虫病肉芽肿的炎症反应并延长宿主存活时间[英]/Mentink-Kane MM,Cheever AW，Thompson RW，et al//Proc Natl Acad Sci U S A , 2005 .

[5]  F. Leyvraz.  Scaling theory and exactly solved models in the kinetics of irreversible aggregation , 2003, cond-mat/0305670.

[6]  Ilya Zaliapin,et al.  A Boolean Delay Equation Model of Colliding Cascades. Part II: Prediction of Critical Transitions , 2003 .

[7]  Jonathan A. D. Wattis,et al.  A Hierarchical Cluster System Based on Horton–Strahler Rules for River Networks , 2002 .

[8]  J. Cardy,et al.  Exact Results for the Universal Area Distribution of Clusters in Percolation, Ising, and Potts Models , 2002, cond-mat/0205404.

[9]  Fausto Guzzetti,et al.  Self-organization, the cascade model, and natural hazards , 2002, Proceedings of the National Academy of Sciences of the United States of America.

[10]  E. Blanter,et al.  Criticality in a dynamic mixed system. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Z. Toroczkai.  Topological classification of binary trees using the Horton-Strahler index. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  M. Newman,et al.  Fast Monte Carlo algorithm for site or bond percolation. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  D. Sornette.  Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools , 2000 .

[14]  Newman,et al.  Critical transitions in colliding cascades , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  D. Turcotte,et al.  Exactly soluble hierarchical clustering model: inverse cascades, self-similarity, and scaling. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  R. Ziff,et al.  SHAPE-DEPENDENT UNIVERSALITY IN PERCOLATION , 1998, cond-mat/9811122.

[17]  D. Sornette,et al.  Log-periodic oscillations for biased diffusion on random lattice , 1997, cond-mat/9712085.

[18]  William I. Newman,et al.  Fractal Trees with Side Branching , 1997 .

[19]  M. G. Shnirman,et al.  Hierarchical model of seismicity: scaling and predictability , 1997 .

[20]  R. Badii,et al.  Complexity: Hierarchical Structures and Scaling in Physics , 1997 .

[21]  L. Botvina,et al.  Premonitory transformation of steel fracturing and seismicity , 1997 .

[22]  Elena Blanter,et al.  SCALING LAWS IN BLOCKS DYNAMICS AND DYNAMIC SELF-ORGANIZED CRITICALITY , 1997 .

[23]  J. Muzy,et al.  Complex fractal dimensions describe the hierarchical structure of diffusion-limited-aggregate clusters. , 1996, Physical review letters.

[24]  Newman,et al.  log-periodic behavior of a hierarchical failure model with applications to precursory seismic activation. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[25]  D. Turcotte.  Fractals and Chaos in Geology and Geophysics , 1992 .

[26]  William I. Newman,et al.  Failure of hierarchical distributions of fibre bundles. I , 1991, International Journal of Fracture.

[27]  L. Knopoff,et al.  Scale invariance in brittle fracture and the dynamics of crack fusion , 1990 .

[28]  D. Stauff,et al.  Monte Carlo and series study of corrections to scaling in two-dimensional percolation , 1984 .

[29]  L. Knopoff,et al.  Crack fusion as a model for repetitive seismicity , 1983 .

[30]  L. Knopoff,et al.  A model for repetitive cycles of large earthquakes , 1983 .

[31]  L. Knopoff,et al.  Crack fusion dynamics: A model for large earthquakes , 1982 .

[32]  A. Provost,et al.  Scaling rules in rock fracture and possible implications for earthquake prediction , 1982, Nature.

[33]  R. J. Harrison,et al.  Monte Carlo experiments on cluster size distribution in percolation , 1979 .

[34]  D. Stauffer.  Violation of Dynamical Scaling for Randomly Dilute Ising Ferromagnets near Percolation Threshold , 1975 .

[35]  Michael E. Fisher,et al.  The theory of condensation and the critical point , 1967 .

[36]  A. N. Strahler.  Quantitative analysis of watershed geomorphology , 1957 .

[37]  B. Gutenberg,et al.  Seismicity of the Earth and associated phenomena , 1950, MAUSAM.

[38]  R. Horton.  EROSIONAL DEVELOPMENT OF STREAMS AND THEIR DRAINAGE BASINS; HYDROPHYSICAL APPROACH TO QUANTITATIVE MORPHOLOGY , 1945 .

[39]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[40]  Yehuda Ben-Zion,et al.  Appendix 2 - Key Formulas in Earthquake Seismology , 2003 .

[41]  D. L. Turcottea,et al.  An inverse-cascade model for self-organized critical behavior , 1999 .

[42]  D L T,et al.  Networks with Side Branching in Biology , 1998 .

[43]  M. G. Shnirman,et al.  Hierarchical model of defect development and seismicity , 1990 .

[44]  Eiji Tokunaga,et al.  Consideration on the composition of drainage networks and their evolution , 1978 .