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# 10 Infinitesimal Generators

[217.3.1] The operators GTαt form a family of strongly continuous semigroups on C0G* provided that the translations Tt inside the integral in eq. (28) are strongly continuous [33, 48] and 0Tshαs/t/tds<. [217.3.2] In this case the infinitesimal generators for 0<α1 are defined by

 Aα⁢ϱ~=s-l⁢i⁢mt→0⁢G⁢Tαt⁢ϱ~-ϱ~t (32)

for all ϱ~C0G* for which the strong limit s-lim exists. [217.3.3] In general, the infinitesimal generators are unbounded operators. [217.3.4] If A=-d/dt denotes the infinitesimal generator of the translation Tt in eq. (28), then

 Aα=--Aα=-dd⁢tα (33)

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

 Aα⁢ϱ~=limϵ→o⁡C⁢∫ϵ∞t-α-1⁢1-Tt⁢ϱ~⁢d⁢t (34)

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

 Aα⁢ϱ~=limϵ→o⁡C⁢∫ϵ∞t-α⁢A⁢1-t⁢A-1⁢ϱ~⁢d⁢t (35)

[page 218, §0]    in terms of the resolvent of A [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 GΓ of small measure in phase space, that are typically incurred in statistical mechanics [1, 21, 50].