[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
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
[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].