Category: Fractional Time

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Ergodicity Fractional Time Irreversibility Statistical Physics Theory of Time

On Local Equilibrium and Ergodicity

R. Hilfer

Acta Physica Polonica B 49, 859 (2018)
DOI: 10.5506/APhysPolB.49.859

submitted on
Friday, April 27, 2018

The main mathematical argument of the universal framework for local equilibrium proposed in Analysis 36, 49 (2016) is condensed and formulated as a fundamental dichotomy between subsets of positive measure and subsets of zero measure in ergodic theory. The physical interpretation of the dichotomy in terms of local equilibria rests on the universality of time scale separation in an appropriate long-time limit.



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Fractional Time Irreversibility Mathematical Physics Nonequilibrium Theory of Time

Mathematical analysis of time flow

R. Hilfer

Analysis 36, 49-64 (2016)
https://doi.org/10.1515/anly-2015-5005

submitted on
Saturday, July 4, 2015

The mathematical analysis of time fow in physical many-body systems leads to the study of long-time limits. This article discusses the interdisciplinary problem of local stationarity, how stationary solutions can remain slowly time dependent after a long-time limit. A mathematical defnition of almost invariant and nearly indistinguishable states on C*-algebras is introduced using functions of bounded mean oscillation. Rescaling of time yields generalized time fows of almost invariant and macroscopically indistinguishable states, that are mathematically related to stable convolution semigroups and fractional calculus. The infnitesimal generator is a fractional derivative of order less than or equal to unity. Applications of the analysis are given to irreversibility and to a physical experiment.



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Ergodic Theory Fractional Calculus Fractional Time Irreversibility Mathematics Theory of Time

Time Automorphisms on C*-Algebras

R. Hilfer

Mathematics 3, 623-643 (2015)
https://doi.org/10.3390/math3030626

submitted on
Tuesday, March 24, 2015

Applications of fractional time derivatives in physics and engineering require the existence of nontranslational time automorphisms on the appropriate algebra of observables. The existence of time automorphisms on commutative and noncommutative C∗-algebras for interacting many-body systems is investigated in this article. A mathematical framework is given to discuss local stationarity in time and the global existence of fractional and nonfractional time automorphisms. The results challenge the concept of time flow as a translation along the orbits and support a more general concept of time flow as a convolution along orbits. Implications for the distinction of reversible and irreversible dynamics are discussed. The generalized concept of time as a convolution reduces to the traditional concept of time translation in a special limit.



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dielectric relaxation Fractional Calculus Fractional Time Theory of Time

Applications and Implications of Fractional Dynamics for Dielectric Relaxation

R. Hilfer

in: Recent Advances in Broadband Dielectric Spectroscopy
edited by: Y. Kalmykov
Springer, Berlin, 123 (2012)
10.1007/978-94-007-5012-8
978-94-007-5011-1

submitted on
Friday, September 23, 2011

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. (eds) Anomalous diffusion: from basis to applications. Springer, Berlin, p 77, 1999; Hilfer, Fractional evolution equations and irreversibility. In: Helbing et al. (eds) Traffic and granular flow’99. Springer, Berlin, p 215, 2000; Hilfer, Fractional time evolution. In: Hilfer (ed) Applications of fractional calculus in physics. World Scientific, Singapore, p 87, 2000; Hilfer, Remarks on fractional time. In: Castell and Ischebeck (eds) Time, quantum and information. Springer, Berlin, p 235, 2003; Hilfer, Physica A 329:35, 2003; Hilfer, Threefold introduction to fractional derivatives. In: Klages et al. (eds) Anomalous transport: foundations and applications. Wiley-VCH, Weinheim, pp 17– 74, 2008; Hilfer, Foundations of fractional dynamics: a short account. In: Klafter et al. (eds) Fractional dynamics: recent advances. World Scientific, Singapore,207, 2011) and demonstrate its relevance and application to broadband dielectric spectroscopy (Hilfer, J Phys Condens Matter 14:2297, 2002; Hilfer, Chem Phys 284:399, 2002; Hilfer, Fractals 11:251, 2003; Hilfer et al., Fractional Calc Appl Anal 12:299, 2009). It was argued, that broadband dielectric spectroscopy might be useful to test effective field theories based on fractional dynamics.



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Fractional Calculus Fractional Time

Foundations of Fractional Dynamics: A Short Account

R. Hilfer

in: Fractional Dynamics: Recent Advances
edited by: J. Klafter and S. Lim and R. Metzler
World Scientific, Singapore, 207 (2011)
https://doi.org/10.1142/8087
ISBN: 978-981-4340-58-8

submitted on
Tuesday, March 22, 2011

Applications of fractional dynamics have received a steadily increasing amount of attention during the past decade. Its foundations have found less interest. This chapter briefly reviews the physical foundations of fractional dynamics.



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Fractional Calculus Fractional Time Theory of Time

Strange Kinetics

R. Hilfer, R. Metzler, A. Blumen, J. Klafter(eds)

Chemical Physics 284, 1 (2002)
https://doi.org/10.1016/S0301-0104(02)00801-7

submitted on
Monday, July 8, 2002

The term strange kinetics originally referred to the dynamics of Hamiltonian systems which, in the limit of weak chaos, display superdiffusion and Levy-walk characteristics. Here we employ the term strange kinetics in a generalized sense to denote all forms of slow kinetics or anomalous dynamics, such as sub-diffusion, superdiffusion, non-Debye relaxation, Levy walks or fractional time evolutions.



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Fractional Calculus Fractional Time Theory of Time

Remarks on Fractional Time

R. Hilfer

in: Time, Quantum and Information
edited by: L. Castell and O. Ischebeck
Springer, Berlin, 235 (2003)
10.1007/978-3-662-10557-3
ISBN: 978-3-540-44033-8

submitted on
Monday, July 1, 2002

It is not possible to repeat an experiment in the past. The underlying philosophical truth in this observation is the difference between certainty of the past and potentiality of the future. This difference is discussed, for example, in C.F.v. Weizsäcker’s papers and it was often pointed out by him in our discussions in the years 1983-1986 in the Starnberg institute. The perennial philosophical problem related to this difference between past and future is the question whether time is real or not.



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dielectric relaxation Fractional Calculus Fractional Time Glasses

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

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

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 Fractional Time Irreversibility Theory of Time

Fractional Evolution Equations and Irreversibility

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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Fractional Calculus Fractional Time Theory of Time

Fractional Time Evolution

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 Time Theory of Time

An Extension of the Dynamical Foundation for the Statistical Equilibrium Concept

R. Hilfer

Physica A 221, 89-96 (1995)
https://doi.org/10.1016/0378-4371(95)00240-8

submitted on
Wednesday, July 19, 1995

This paper reviews a recently introduced generalization of dynamical stationarity involving the appearance of stable convolution semigroups in the ultralong time limit. Dynamical stationarity is the basis of the equilibrium concept in statistical mechanics, and the ultralong time limit is a limit in which a discretized time flow is iterated infinitely often while the discretization time step becomes infinite. The new limit is necessary when investigating induced automorphisms for subsets of measure zero. It is found that the induced dynamics of subsets of zero measure is given generically by stable convolution semigroups and not by the conventional translation group. This could provide insight into the macroscopic irreversibility paradox. The induced semigroups are generated by fractional time derivatives of orders less than unity, not by a first-order time derivative as the conventional group. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractional stationarity provides the dynamical foundation for a generalized equilibrium concept.



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Fractional Time Theory of Time

Foundations of Fractional Dynamics

R. Hilfer

Fractals 3, 549 (1995)
https://doi.org/10.1142/S0218348X95000485

submitted on
Monday, March 6, 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 by fractional time derivatives of orders less than unity, and not by a first order time derivative. Invariance under the induced semiflows therefore leads to a new form of stationarity, called fractional stationarity. Fractionally stationary states are dissipative. Fractional stationarity also provides the dynamical foundation for a previously proposed generalized equilibrium concept.



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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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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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