Category: Fractional Calculus

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

Post author By R Hilfer
Post date December 7, 2001

R. Hilfer

Chem.Phys. 284, 399 (2002)
https://doi.org/10.1016/S0301-0104(02)00670-5

submitted on
Friday, December 7, 2001

The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless it contains only a single stretching exponent. Qualitative and quantitative agreement with dielectric data for glycerol is found that extends into the excess wing. The fits require fewer parameters than traditional fit functions and can extend over up to 13 decades in frequency.

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dielectric relaxation Fractional Calculus Fractional Time Glasses

Fitting the excess wing in the dielectric α-relaxation of propylene carbonate

Post author By R Hilfer
Post date November 28, 2001

R. Hilfer

Journal of Physics: Condensed Matter 14, 2297 (2002)
https://doi.org/10.1088/0953-8984/14/9/318

submitted on
Wednesday, November 28, 2001

A novel fitting function for the complex frequency-dependent dielectric susceptibility is introduced and compared against other fitting functions for experimental broadband dielectric loss spectra of propylene carbonate taken from Schneider et al (Schneider U, Lunkenheimer P, Brand R and Loidl A 1999 Phys. Rev. E 59 6924). The fitting function contains a single stretching exponent similar to the familiar Cole–Davidson or Kohlrausch stretched exponential fits. It is compared to these traditional fits as well as to the Havriliak–Negami susceptibility and a susceptibility for a two-step Debye relaxation. The results for the novel fit are found to give superior agreement.

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Fractional Calculus Glasses

On Fractional Relaxation

Post author By R Hilfer
Post date April 2, 2001

R. Hilfer

Fractals 11, 251 (2003)
https://doi.org/10.1142/S0218348X03001914

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β -relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.

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Fractional Calculus Glasses

On Fractional Relaxation

Post author By R Hilfer
Post date April 2, 2001

R. Hilfer

in: Scaling and Disordered Systems
edited by: F. Family and M. Daoud and H. Herrmann and H.E. Stanley
World Scientific, Singapore, 251 (2002)
https://doi.org/10.1142/9789812778109_0026
ISBN: 978-981-02-4838-3

submitted on
Monday, April 2, 2001

Generalized fractional relaxation equations based on generalized Riemann-Liouville derivatives are combined with a simple short time regularization and solved exactly. The solution involves generalized Mittag-Leffler functions. The associated frequency dependent susceptibilities are related to symmetrically broadened Cole-Cole susceptibilities occurring as Johari Goldstein β-relaxation in many glass formers. The generalized susceptibilities exhibit a high frequency wing and strong minimum enhancement.

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diffusion Fractional Calculus

Fractional Diffusion based on Riemann-Liouville Fractional Derivatives

Post author By R Hilfer
Post date October 12, 1999

R. Hilfer

The Journal of Physical Chemistry B 104, 3914-3917 (2000)
DOI: 10.1021/jp9936289

submitted on
Tuesday, October 12, 1999

A fractional diffusion equation based on Riemann−Liouville fractional derivatives is solved exactly. The initial values are given as fractional integrals. The solution is obtained in terms of H-functions. It differs from the known solution of fractional diffusion equations based on fractional integrals. The solution of fractional diffusion based on a Riemann−Liouville fractional time derivative does not admit a probabilistic interpretation in contrast with fractional diffusion based on fractional integrals. While the fractional initial value problem is well defined and the solution finite at all times, its values for t → 0 are divergent.

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Tags Anomalous Diffusion

Fractional Calculus Fractional Time Irreversibility Theory of Time

Fractional Evolution Equations and Irreversibility

Post author By R Hilfer
Post date September 27, 1999

R. Hilfer

in: Traffic and Granular Flow’99
edited by: D. Helbing and H. Herrmann and M. Schreckenberg and D. Wolf
Springer, Berlin, 215 (2000)
10.1007/978-3-642-59751-0
ISBN: 978-3-642-64109-1

submitted on
Monday, September 27, 1999

The paper reviews a general theory predicting the general importance of fractional evolution equations. Fractional time evolutions are shown to arise from a microscopic time evolution in a certain long time scaling limit. Fractional time evolutions are generally irreversible. The infinitesimal generators of fractional time evolutions are fractional time derivatives. Evolution equations containing fractional time derivatives are proposed for physical, economical and traffic applications. Regular non-fractional time evolutions emerge as special cases from the results. Also for these regular time evolutions it is found that macroscopic irreversibility arises in the scaling limit.

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Equilibrium Fractional Calculus Statistical Physics

Fractional Calculus and Regular Variation in Thermodynamics

Post author By R Hilfer
Post date May 5, 1999

R. Hilfer

in: Applications of Fractional Calculus in Physics
edited by: R. Hilfer
World Scientific, Singapore, 429-463 (2000)
https://doi.org/10.1142/3779
ISBN: 978-981-02-3457-7

submitted on
Wednesday, May 5, 1999

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Tags regular variation, scaling

Fractional Calculus Fractional Time Theory of Time

Fractional Time Evolution

Post author By R Hilfer
Post date May 5, 1999

R. Hilfer

in: Applications of Fractional Calculus in Physics
edited by: R. Hilfer
World Scientific, Singapore, 87-130 (2000)
https://doi.org/10.1142/3779
ISBN: 978-981-02-3457-7

submitted on
Wednesday, May 5, 1999

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Fractional Calculus

Applications of Fractional Calculus in Physics

Post author By R Hilfer
Post date May 5, 1999

R. Hilfer

World Scientific Publ. Co., Singapore, 2000
https://doi.org/10.1142/3779
ISBN: 978-981-02-3457-7

submitted on
Wednesday, May 5, 1999

Although fractional calculus is a natural generalization of calculus, and although its mathematical history is equally long, it has, until recently, played a negligible role in physics. One reason could be that, until recently, the basic facts were not readily accessible even in the mathematical literature. This book intends to increase the accessibility of fractional calculus by combining an introduction to the mathematics with a review of selected recent applications in physics.

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diffusion Fractional Calculus Random Walks

On Fractional Diffusion and its Relation with Continuous Time Random Walks

Post author By R Hilfer
Post date May 22, 1998

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 review article

Fractional Derivatives in Static and Dynamic Scaling

Post author By R Hilfer
Post date March 11, 1997

R. Hilfer

in: Scale Invariance and Beyond
edited by: B. Dubrulle and F. Graner and D. Sornette
Springer, Berlin, 53 (1997)
10.1007/978-3-662-09799-1
978-3-540-64000-4

submitted on
Tuesday, March 11, 1997

The paper is a brief review of recent applications of fractional calculus in physics with emphasis on static and dynamic scaling.

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Tags scaling

Fractional Calculus Fractional Time Random Walks Stochastic Processes Theory of Time

Fractional Master Equations and Fractal Time Random Walks

Post author By R Hilfer
Post date October 28, 1994

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

Post author By R Hilfer
Post date October 20, 1994

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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