[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
![]() |
|||
![]() |
(29) |
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
.