Tag: CTRW

Categories
Stochastic Processes

Composite Continuous Time Random Walks

R. Hilfer

Eur.Phys.J. B 90, 233 (2017)
https://doi.org/10.1140/epjb/e2017-80369-y

submitted on
Saturday, May 20, 2017

Random walks in composite continuous time are introduced. Composite time flow is the product of translational time flow and fractional time flow [see Chem. Phys. 84, 399 (2002)]. The continuum limit of composite continuous time random walks gives a diffusion equation where the infinitesimal generator of time flow is the sum of a first order and a fractional time derivative. The latter is specified as a generalized Riemann-Liouville derivative. Generalized and binomial Mittag-Leffler functions are found as the exact results for waiting time density and mean square displacement.



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Categories
diffusion Fractional Calculus Random Walks

On Fractional Diffusion and its Relation with Continuous Time Random Walks

R. Hilfer

in: Anomalous Diffusion: From Basis to Applications
edited by: R. Kutner, A. Pekalski and K. Sznajd-Weron
Lecture Notes in Physics, vol. 519,Springer, Berlin, 77 (1999)
10.1007/BFb0106828
978-3-662-14242-4

submitted on
Friday, May 22, 1998

Time evolutions whose infinitesimal generator is a fractional time derivative arise generally in the long time limit. Such fractional time evolutions are considered here for random walks. An exact relationship is established between the fractional master equation and a separable continuous time random walk of the Montroll-Weiss type. The waiting time density can be expressed using a generalized Mittag-Leffier function. The first moment of the waiting density does not exist.



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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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Categories
Disordered Systems electrical conductivity Percolation Random Walks Stochastic Processes Transport Processes

Continuous Time Random Walk Approach to Dynamic Percolation

R. Hilfer, R. Orbach

Chemical Physics 128, 275 (1988)
https://doi.org/10.1016/0301-0104(88)85076-6

submitted on
Friday, September 16, 1988

We present an approximate solution for time (frequency) dependent response under conditions of dynamic percolation which may be related to excitation transfer in some random structures. In particular, we investigate the dynamics of structures where one random component blocks a second (carrier) component. Finite concentrations of the former create a percolation network for the latter. When the blockers are allowed to move in time, the network seen by the carriers changes with time, allowing for long-range transport even if the instantaneous carrier site availability is less than pc, the critical percolation concentration. A specific example of this situation is electrical transport in sodium β”-alumina. The carriersare Na+ ions which can hop on a two-dimensional honeycomb lattice. The blockers are ions of much higher activation energy, such as Ba++. We study the frequency dependence of the conductivity for such a system. Given a fixed Ba++ hopping rate the Na+ ions experience a frozen site percolation environment for frequencies larger than the inverse hopping rate. At frequencies smaller than the inverse hopping rate, the Na+ ions experience a dynamic environment which allows long-rangetransport, even below the percoltion threshold. A continuous time random walk mode1 combined with an effective medium approximation allows us to arrive at a numerical solution for the frequency-dependent Na+ conductivity which clearly exhibits the crossover from frozen to dynamic environment.



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