Category: Irreversibility

On Local Equilibrium and Ergodicity

Post author By R Hilfer
Post date April 27, 2018

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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Tags 2018, ergodicity breaking, local equilibrium, Scaling Limit

Fractional Time Irreversibility Mathematical Physics Nonequilibrium Theory of Time

Mathematical analysis of time flow

Post author By R Hilfer
Post date July 4, 2015

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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Tags 2016, C*-algebras, Scaling Limit

Ergodic Theory Fractional Calculus Fractional Time Irreversibility Mathematics Theory of Time

Time Automorphisms on C*-Algebras

Post author By R Hilfer
Post date March 24, 2015

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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Tags 2015, C*-algebras

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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