[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].