Ergodic Theory – R. Hilfer

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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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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