[217.3.1] The operators form a family of strongly continuous semigroups on provided that the translations inside the integral in eq. (28) are strongly continuous [33, 48] and . [217.3.2] In this case the infinitesimal generators for are defined by

(32) |

for all for which the strong limit exists. [217.3.3] In general, the infinitesimal generators are unbounded operators. [217.3.4] If denotes the infinitesimal generator of the translation in eq. (28), then

(33) |

are fractional time derivatives [50, 16]. [217.3.5] The action of on mixed states can be represented in different ways. [217.3.6] Frequently an integral representation

(34) |

of Marchaud type [8, 51] is used. [217.3.7] The integral representation

(35) |

[page 218, §0] in terms of the resolvent of [12] defines the same fractional derivative operator [52]. [218.0.1] Representations of Grünwald-Letnikov type are also well known [16].

[218.1.1] In summary, fractional dynamical systems must be expected to appear generally in mathematical models of macroscopic phenomena. [218.1.2] They arise as coarse grained macroscopic time evolutions from inducing a microscopic time evolution on the subsets of small measure in phase space, that are typically incurred in statistical mechanics [1, 21, 50].