6 Composite fractional time evolutions

[403.2.3.1] In the previous section it was mentioned that the transition from microscopic to macroscopic time scales leads to the replacement T1⁢t→Tα⁢t[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, T1 and Tα, 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 T1 and a fractional translation Tα

where 0<ε=τ2/τ1<∞ is the ratio of time scales. [403.2.3.4] T~α is called a composite fractional time evolution of order α. [403.2.3.5] For ε=1 translation T1⁢t [page 404, §0]    and fractional time evolution Tα⁢t occurr simultaneously on the same time scale. [404.1.0.1] For ε→0 the standard translation results while for ε→∞ the combined time evolution approaches a fractional translation.

[404.1.1.1] First note that with g⁢t0=Tα⁢t2⁢f⁢t0 and for any admissible function f

it follows that T1 and Tα commute. [404.1.1.2] Next observe that T~α is again a semi-group because

obeys the semi-group relation by virtue of eq. (6).

[404.1.2.1] The infinitesimal generator A~α=limt→0+⁡T~α⁢t-1/t of composite fractional translations is calculated as

where A=-d/d⁢t is the infinitesimal generator of T1⁢t and Aα, the infinitesimal generators of Tα⁢t, is the Marchaud-Hadamard fractional derivative [3].

[404.1.3.1] These considerations suggest to replace the time evolution T1⁢t in a microscopic equation of motion with T~α⁢t. [404.1.3.2] As a consequence the infinitesimal generator d/d⁢t of time evolution has to be replaced with the generator A~α of composite fractional translations. [404.1.3.3] Possible generalizations of composite fractional time evolutions may be obtained by generalizing T~α⁢t into T~α1,α2⁢t=Tα1⁢t⁢Tα2⁢t. [404.1.3.4] Further generalization is possible by iterating the replacement to get T~α1,α2,…,αn=T~α1,α2,…,αn-1⁢Tαn⁢t.