Extreme events as foundation of Lévy walks with varying velocity

Abstract

Abstract In this work we study the role of extreme events [E.W. Montroll, B.J. West, in: J.L. Lebowitz, E.W. Montrell (Eds.), Fluctuation Phenomena, SSM, vol. VII, North-Holland, Amsterdam, 1979, p. 63; J.-P. Bouchaud, M. Potters, Theory of Financial Risks from Statistical Physics to Risk Management, Cambridge University Press, Cambridge, 2001; D. Sornette, Critical Phenomena in Natural Sciences. Chaos, Fractals, Selforganization and Disorder: Concepts and Tools, Springer, Berlin, 2000] in determining the scaling properties of Levy walks with varying velocity. This model is an extension of the well-known Levy walks one [J. Klafter, G. Zumofen, M.F. Shlesinger, in M.F. Shlesinger, G.M. Zaslavsky, U. Frisch (Eds.), Levy Flights and Related Topics ion Physics, Lecture Notes in Physics, vol. 450, Springer, Berlin, 1995, p. 196; G. Zumofen, J. Klafter, M.F. Shlesinger, in: R. Kutner, A. Pȩkalski, K. Sznajd-Weron (Eds.), Anomalous Diffusion. From Basics to Applications, Lecture Note in Physics, vol. 519, Springer, Berlin, 1999, p. 15] introduced in the context of chaotic dynamics where a fixed value of the walker velocity is assumed for simplicity. Such an extension seems to be necessary when the open and/or complex system is studied. The model of Levy walks with varying velocity is spanned on two coupled velocity–temporal hierarchies: the first one consisting of velocities and the second of corresponding time intervals which the walker spends between the successive turning points. Both these hierarchical structures are characterized by their own self-similar dimensions. The extreme event, which can appear within a given time interval, is defined as a single random step of the walker having largest length. By finding power-laws which describe the time-dependence of this displacement and its statistics we obtained two independent diffusion exponents, which are related to the above-mentioned dimensions and which characterize the extreme event kinetics. In this work we show the principal influence of extreme events on the basic quantities (one-step distributions and moments as well as two-step correlation functions) of the continuous-time random walk formalism. Besides, we construct both the waiting-time distribution and sojourn probability density directly in a real space and time in the scaling form by proper component analysis which takes into account all possible fluctuations of the walker steps in contrast to the extreme event analysis. In this work we pay our attention to the basic quantities, since the summarized multi-step ones were already discussed earlier [Physica A 264 (1999) 107; Comp. Phys. Commun. 147 (2002) 565]. Moreover, we study not only the scaling phenomena but also, assuming a finite number of hierarchy levels, the breaking of scaling and its dependence on control parameters. This seems to be important for studying empirical systems the more so as there are still no closed formulae describing this phenomenon except the one for truncated Levy flights [Phys. Rev. Lett. 73 (1994) 2946]. Our formulation of the model made possible to develop an efficient Monte Carlo algorithm [Physica A 264 (1999) 107; Comp. Phys. Commun. 147 (2002) 565] where no MC step is lost.