Abstract
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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References
In classical mechanics the states are points in phase space, the observables are functions on phase space, and the operator B is specified by a vector field and Poisson brackets. In quantum mechanics (with finitely many degrees of freedom) the states correspond to rays in a Hilbert space, the observables to operators on this space, and the operator B to the Hamiltonian. In field theories the states are normalized positive functionals on an operator algebra of observables, and then B becomes a derivation on the algebra of observables. The equations (1) need not be first order in time. An example is the initial-value problem for the wave equation for g(t,x) 2???9 dt2 C dx2 in one dimension. It can be recast into the form of (1) by introducing a second variable h and defining ’ - ( 0 - » - (??? jk
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Hilfer, R. (2000). Fractional Evolution Equations and Irreversibility. In: Helbing, D., Herrmann, H.J., Schreckenberg, M., Wolf, D.E. (eds) Traffic and Granular Flow ’99. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-59751-0_20
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DOI: https://doi.org/10.1007/978-3-642-59751-0_20
Publisher Name: Springer, Berlin, Heidelberg
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