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## Mathematical and physical interpretations of fractional derivatives and integrals

### R. Hilfer

in: Handbook of Fractional Calculus with Applications: Basic Theory, Vol. 1
edited by: A. Kochubei and Y. Luchko
Walter de Gruyter GmbH, Berlin, 47-86 (2019)
https://doi.org/10.1515/9783110571622
ISBN: 9783110571622

submitted on
Saturday, June 2, 2018

Brief descriptions of various mathematical and physical interpretations of fractional derivatives and integrals have been collected into this chapter as points of reference and departure for deeper studies. “Mathematical interpretation” in the title means a brief description of the basic mathematical idea underlying a precise definition. “Physical interpretation” means a brief description of the physical theory underlying an identification of the fractional order with a known physical quantity. Numerous interpretations had to be left out due to page limitations. Only a crude, rough and ready description is given for each interpretation. For precise theorems and proofs an extensive list of references can serve as a starting point.

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## 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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## 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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## Statistical Prediction of Corrosion Front Penetration

### T. Johnsen, R. Hilfer

Phys.Rev. E 55, 5433 (1997)
https://doi.org/10.1103/PhysRevE.55.5433

submitted on
Wednesday, September 18, 1996

A statistical method to predict the stochastic evolution of corrosion fronts has been developed. The method is based on recording material loss and maximum front depth. In this paper we introduce the method and test its applicability. In the absence of experimental data we use simulation data from a three-dimensional corrosion model for this test. The corrosion model simulates localized breakdown of a protective oxide layer, hydrolysis of corrosion product and repassivation of the exposed surface. In the long time limit of the model, pits tend to coalesce. For different model parameters the model reproduces corrosion patterns observed in experiment. The statistical prediction method is based in the theory of stochastic processes. It allows the estimation of conditional probability densities for penetration depth, pitting factor, residual lifetimes, and corrosion rates which are of technological interest.

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## 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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## 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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## 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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## 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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## Fluctuation-Dissipation on Fractals: A Probabilistic Approach

### R. Hilfer, A. Blumen

in: Time Dependent Effects in Disordered Materials
edited by: R.Pynn and T. Riste
Plenum Press, New York, 217 (1987)

submitted on
Tuesday, March 31, 1987

The analogies between the diffusion problem and the resistor network problem as witnessed by the Einstein relation have been very important for analytical and numerical investigations of linear problems in disordered geometries (e.g. percolating clusters). This raises the question whether the resistor problem can be identified in a purely probabilistic context. An affirmative answer has recently been given and it was shown that the Einstein relation follows from a simple probabilistic argument. Here we present the results of a more general treatment.

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## Renormalisation Group Approach in the Theory of Disordered Systems

### R. Hilfer

Verlag Harri Deutsch, Frankfurt, 1986
ISBN-10: 3871449792, ISBN-13: 978-3871449796

submitted on
Wednesday, July 23, 1986

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## On Finitely Ramified Fractals and Their Extensions

### R. Hilfer, A. Blumen

in: Fractals in Physics
edited by: L. Pietronero and E. Tosatti
Elsevier Publishing Co., Amsterdam, 33 (1986)

submitted on
Thursday, July 11, 1985

We construct deterministic fractal lattices using generators with tetrahedral symmetry. From the corresponding master equation we determine the spectral dimension d and prove that d<2. Furthermore we extend our set of fractals (with d dense in [1,2]) by direct multiplication, thus obtaining fractals whose d are dense for all real numbers larger than or equal to 1 .

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## Stochastic Models for Corporate Planning

### R. Hilfer

GBI — Verlag, München, 1985
ISBN-10: 3890030157, ISBN-13: 978-3890030159

submitted on
Wednesday, December 5, 1984

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## Renormalisation on Symmetric Fractals

### R. Hilfer, A. Blumen

J.Phys.A: Math. Gen. 17, L783 (1984)
https://doi.org/10.1088/0305-4470/17/14/011

submitted on
Monday, July 9, 1984

We introduce and investigate new classes of Sierpinski-type fractals. We determine their fractal and spectral dimensions using renormalisation procedures and, for particular classes, we give these dimensions in closed form. The spectral dimensions densely fill the interval [1,2], allowing us to choose flexibly models for applications.

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## Renormalisation on Sierpinski-type Fractals

### R. Hilfer, A. Blumen

Journal of Physics A: Mathematical and General 17, L573-L545 (1984)
10.1088/0305-4470/17/10/004

submitted on
Friday, April 13, 1984

We present a family of deterministic fractals which generalise the d-dimensional Sierpinski gaskets and we establish their order of ramification and their fractal and spectral dimensions. Random walks on these fractals are renormalisable and lead to rational, not necessarily polynomial, mappings.