[403.2.3.1] In the previous section it was mentioned that the transition from microscopic to macroscopic time scales leads to the replacement [3]. [403.2.3.2] In nature the ratio of microscopic to macroscopic time scales may be small but is never exactly zero, and one expects that both time evolutions, and , are simultaneously present when the ratio is finite. [403.2.3.3] Therefore it becomes of interest to study also a composite time evolution consisting of a simple shift and a fractional translation

(28) |

where is the ratio of time scales. [403.2.3.4] is called a composite fractional time evolution of order . [403.2.3.5] For translation [page 404, §0] and fractional time evolution occurr simultaneously on the same time scale. [404.1.0.1] For the standard translation results while for the combined time evolution approaches a fractional translation.

[404.1.1.1] First note that with and for any admissible function

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it follows that and commute. [404.1.1.2] Next observe that is again a semi-group because

(30) |

obeys the semi-group relation by virtue of eq. (6).

[404.1.2.1] The infinitesimal generator of composite fractional translations is calculated as

(31) |

where is the infinitesimal generator of and , the infinitesimal generators of , is the Marchaud-Hadamard fractional derivative [3].

[404.1.3.1] These considerations suggest to replace the time evolution in a microscopic equation of motion with . [404.1.3.2] As a consequence the infinitesimal generator of time evolution has to be replaced with the generator of composite fractional translations. [404.1.3.3] Possible generalizations of composite fractional time evolutions may be obtained by generalizing into . [404.1.3.4] Further generalization is possible by iterating the replacement to get .