Search results for "Fractional dynamics"
showing 3 items of 3 documents
FOUNDATIONS OF FRACTIONAL DYNAMICS
1995
Time flow in dynamical systems is reconsidered in the ultralong time limit. The ultralong time limit is a limit in which a discretized time flow is iterated infinitely often and the discretization time step is infinite. The new limit is used to study induced flows in ergodic theory, in particular for subsets of measure zero. Induced flows on subsets of measure zero require an infinite renormalization of time in the ultralong time limit. It is found that induced flows are given generically by stable convolution semigroups and not by the conventional translation groups. This could give new insight into the origin of macroscopic irreversibility. Moreover, the induced semigroups are generated …
Applications and Implications of Fractional Dynamics for Dielectric Relaxation
2012
This article summarizes briefly the presentation given by the author at the NATO Advanced Research Workshop on “Broadband Dielectric Spectroscopy and its Advanced Technological Applications”, held in Perpignan, France, in September 2011. The purpose of the invited presentation at the workshop was to review and summarize the basic theory of fractional dynamics (Hilfer, Phys Rev E 48:2466, 1993; Hilfer and Anton, Phys Rev E Rapid Commun 51:R848, 1995; Hilfer, Fractals 3(1):211, 1995; Hilfer, Chaos Solitons Fractals 5:1475, 1995; Hilfer, Fractals 3:549, 1995; Hilfer, Physica A 221:89, 1995; Hilfer, On fractional diffusion and its relation with continuous time random walks. In: Pekalski et al. …
Nonlocal Fractional Dynamics for Different Terminal Densities
2018
We study the effect of confining potentials, generated by different equilibrium (long-time asymptotic or terminal) probability densities, on nonGaussian stochastic processes, described by Lévy–Schrödinger semigroup dynamics. The former densities belong to the family of so-called M-Wright functions of index ν. Using analytical and numerical arguments, we demonstrate that properly tailored confining potentials can generate the Gaussian distribution (which is also a member of M-Wright family at ν = 1/2) at final stages of time evolution. This means that the Gaussian distribution (and other sufficiently fast decaying distributions like exponential one) can emerge in the differential equation wi…