The stochastic nature of complexity evolution in the fractional systems

Abstract The stochastic scenario of relaxation in the complex systems is presented. It is based on a general probabilistic formalism of limit theorems. The nonexponential relaxation is shown to result from the asymptotic self-similar properties in the temporal behavior of such systems. This model provides a rigorous justification of the energy criterion introduced by Jonscher. The meaning of the parameters into the empirical response functions is clarified. This treatment sheds a fresh light on the nature of not only the dielectric relaxation but also mechanical, luminescent and radiochemical ones. In the case of the Cole–Cole response there exists a direct link between the notation of the fractional derivative (appearing in the fractional macroscopic equation often proposed) and the model. But the macroscopic response equations, relating to the Cole–Davidson and Havriliak–Negami relaxations, have a more general integro-differential form in comparison with the ordinary fractional one.

[1]  Karina Weron,et al.  A relationship between asymmetric Lévy-stable distributions and the dielectric susceptibility , 1993 .

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

[3]  R. M. Hill,et al.  The fractal nature of the cluster model dielectric response functions , 1989 .

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

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

[6]  Yonezawa,et al.  Anomalous relaxation in the fractal time random walk model. , 1995, Physical review letters.

[7]  G. Niklasson.  A fractal description of the dielectric response of disordered materials , 1993 .

[8]  Thermodynamics and fractional Fokker-Planck equations. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  K. Weron,et al.  On the Cole-Cole relaxation function and related Mittag-Leffler distribution , 1996 .

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

[11]  P. Lee,et al.  14. Simulation and Chaotic Behaviour of α‐Stable Stochastic Processes , 1995 .

[12]  Michele Caputo,et al.  A model for the fatigue in elastic materials with frequency independent Q , 1979 .

[13]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[14]  Yonezawa,et al.  Anomalous relaxation in fractal structures. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[15]  Miss A.O. Penney.  (b) , 1974, The New Yale Book of Quotations.

[16]  M. R. D. L. Fuente,et al.  The memory effect in dielectric response , 1985 .

[17]  Stochastic Schemes of Dielectric Relaxation in Correlated-Cluster Systems , 2002, cond-mat/0210481.

[18]  R. N. Pillai.  On Mittag-Leffler functions and related distributions , 1990 .

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

[20]  J Klafter,et al.  On the relationship among three theories of relaxation in disordered systems. , 1986, Proceedings of the National Academy of Sciences of the United States of America.

[21]  F. Mainardi,et al.  Fractals and fractional calculus in continuum mechanics , 1997 .

[22]  K. Weron,et al.  On the equivalence of the parallel channel and the correlated cluster relaxation models , 1997 .

[23]  A. Jonscher.  Dielectric relaxation in solids , 1983 .

[24]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[25]  W. R. Buckland,et al.  Distributions in Statistics: Continuous Multivariate Distributions , 1974 .

[26]  A. Jurlewicz.  Limit theorems for randomly coarse grained continuous-time random walks , 2005 .

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

[28]  B. Gnedenko,et al.  Limit Distributions for Sums of Independent Random Variables , 1955 .

[29]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[30]  K. Weron,et al.  Random Walk Approach to Relaxation in Disordered Systems , 1996 .

[31]  A. Stanislavsky,et al.  Memory effects and macroscopic manifestation of randomness. , 2000, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[32]  Aleksander Stanislavsky,et al.  Probability Interpretation of the Integral of Fractional Order , 2004 .

[33]  R. Gorenflo,et al.  Fractional Calculus: Integral and Differential Equations of Fractional Order , 2008, 0805.3823.

[34]  K. Weron,et al.  Energy criterion in interacting cluster systems , 2001 .

[35]  Raoul R. Nigmatullin,et al.  Cole-Davidson dielectric relaxation as a self-similar relaxation process , 1997 .

[36]  Aleksander Stanislavsky,et al.  Exact solution of averaging procedure over the Cantor set , 2002 .

[37]  W. R. Buckland,et al.  Distributions in Statistics: Continuous Multivariate Distributions , 1973 .

[38]  DeGuire Mr,et al.  Two-level model for the nonexponential Williams-Watts dielectric relaxation. , 1988 .

[39]  A. Stanislavsky.  Fractional dynamics from the ordinary Langevin equation. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  P. Grigolini,et al.  Fractional calculus as a macroscopic manifestation of randomness , 1999 .

[41]  A. Weron,et al.  Simulation and Chaotic Behavior of Alpha-stable Stochastic Processes , 1993 .

[42]  K. Weron,et al.  Probabilistic basis for the Cole-Cole relaxation law , 2000 .