Category: Mathematics

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Fractional Calculus Fractional Time Random Walks Stochastic Processes Theory of Time

Fractional Master Equations and Fractal Time Random Walks

R. Hilfer, L. Anton

Physical Review E, Rapid Communication 51, R848 (1995)
https://doi.org/10.1103/PhysRevE.51.R848

submitted on
Friday, October 28, 1994

Fractional master equations containing fractional time derivatives of order less than one are introduced on the basis of a recent classification of time generators in ergodic theory. It is shown that fractional master equations are contained as a special case within the traditional theory of continuous time random walks. The corresponding waiting time density is obtained exactly in terms of the generalized Mittag-Leffler function. This waiting time distribution is singular both in the long time as well as in the short time limit.



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Fractional Calculus Stochastic Processes

Exact Solutions for a Class of Fractal Time Random Walks

R. Hilfer

Fractals 3(1), 211-216 (1995)
https://doi.org/10.1142/S0218348X95000163

submitted on
Thursday, October 20, 1994

Fractal time random walks with generalized Mittag-Leffler functions as waiting time densities are studied. This class of fractal time processes is characterized by a dynamical critical exponent 0<ω≤1, and is equivalently described by a fractional master equation with time derivative of noninteger order ω. Exact Greens functions corresponding to fractional diffusion are obtained using Mellin transform techniques. The Greens functions are expressible in terms of general H-functions. For ω<1 they are singular at the origin and exhibit a stretched Gaussian form at infinity. Changing the order ω interpolates smoothly between ordinary diffusion ω=1 and completely localized behavior in the ω→0 limit.



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Ergodic Theory Ergodicity Fractional Time Mathematical Physics Theory of Time

Fractional Dynamics, Irreversibility and Ergodicity Breaking

R. Hilfer

Chaos, Solitons and Fractals 5, 1475 (1995)
https://doi.org/10.1016/0960-0779(95)00027-2

submitted on
Wednesday, September 28, 1994

Time flow in dynamical systems is analysed within the framework of ergodic theory from the perspective of a recent classification theory of phase transitions. Induced automorphisms are studied on subsets of measure zero. The induced transformations are found to be stable convolution semigroups rather than translation groups. This implies non-uniform flow of time, time irreversibility and ergodicity breaking. The induced semigroups are generated by fractional time derivatives. Stationary states with respect to fractional dynamics are dissipative in the sense that the measure of regions in phase space may decay algebraically with time although the measure is time transformation invariant.



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Fractals Mathematics Stochastic Processes Transport Processes

Probabilistic Interpretation of the Einstein Relation

R. Hilfer, A. Blumen

Physical Review A 37, 578 (1988)
10.1103/PhysRevA.37.578

submitted on
Monday, June 8, 1987

We present a probabilistic picture for the Einstein relation which holds for arbitrarily connected structures. The diffusivity is related to mean first-passage times, while the conductance is given as a direct-passage probability. The fractal Einstein relation is an immediate consequence of our result. In addition, we derive a star-triangle transformation for Markov chains and calculate the exact values of the fracton (spectral) dimension for treelike structures. We point to the relevance of the probabilistic interpretation for simulation and experiment.



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