Fractional Time Theory of Time

An Extension of the Dynamical Foundation for the Statistical Equilibrium Concept

R. Hilfer

Physica A 221, 89-96 (1995)

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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