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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 T_{1}(t)\to T_{\alpha}(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, T_{1} and T_{\alpha}, 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 T_{1} and a fractional translation T_{\alpha}

\tilde{T}_{\alpha}(\tau _{1}t)=T_{1}(\tau _{1}t)T_{\alpha}(\tau _{2}t)=T_{1}(\tau _{1}t)T_{\alpha}(\tau _{1}\varepsilon t) (28)

where 0<\varepsilon=\tau _{2}/\tau _{1}<\infty is the ratio of time scales. [403.2.3.4] \tilde{T}_{\alpha} is called a composite fractional time evolution of order \alpha. [403.2.3.5] For \varepsilon=1 translation T_{1}(t) [page 404, §0]    and fractional time evolution T_{\alpha}(t) occurr simultaneously on the same time scale. [404.1.0.1] For \varepsilon\to 0 the standard translation results while for \varepsilon\to\infty the combined time evolution approaches a fractional translation.

[404.1.1.1] First note that with g(t_{0})=(T_{\alpha}(t_{2})f)(t_{0}) and for any admissible function f

\displaystyle=\int _{0}^{\infty}f(t_{0}-t_{1}-s)h_{\alpha}\left(\frac{s}{t_{2}}\right)\frac{\mathrm{d}s}{t_{2}}=\int _{0}^{\infty}\left(T_{1}(t_{1})f\right)(t_{0}-s)h_{\alpha}\left(\frac{s}{t_{2}}\right)\frac{\mathrm{d}s}{t_{2}}=\left(T_{\alpha}(t_{2})\left(T_{1}(t_{1})f\right)\right)(t_{0}) (29)

it follows that T_{1} and T_{\alpha} commute. [404.1.1.2] Next observe that \tilde{T}_{\alpha} is again a semi-group because

\tilde{T}_{\alpha}(t_{1}+t_{2})=T_{1}(t_{1}+t_{2})T_{\alpha}(t_{1}+t_{2})=T_{1}(t_{1})T_{1}(t_{2})T_{\alpha}(t_{1})T_{\alpha}(t_{2})=T_{1}(t_{1})T_{\alpha}(t_{1})T_{1}(t_{2})T_{\alpha}(t_{2})=\tilde{T}_{\alpha}(t_{1})\tilde{T}_{\alpha}(t_{2}) (30)

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

[404.1.2.1] The infinitesimal generator \tilde{A}_{\alpha}=\lim _{{t\to 0+}}\linebreak(\tilde{T}_{\alpha}(t)-1)/t of composite fractional translations is calculated as

\tilde{A}_{\alpha}=A+A_{\alpha} (31)

where A=-\mathrm{d}/\mathrm{d}t is the infinitesimal generator of T_{1}(t) and A_{\alpha}, the infinitesimal generators of T_{\alpha}(t), is the Marchaud-Hadamard fractional derivative [3].

[404.1.3.1] These considerations suggest to replace the time evolution T_{1}(t) in a microscopic equation of motion with \tilde{T}_{\alpha}(t). [404.1.3.2] As a consequence the infinitesimal generator \mathrm{d}/\mathrm{d}t of time evolution has to be replaced with the generator \tilde{A}_{\alpha} of composite fractional translations. [404.1.3.3] Possible generalizations of composite fractional time evolutions may be obtained by generalizing \tilde{T}_{\alpha}(t) into \tilde{T}_{{\alpha _{1},\alpha _{2}}}(t)=T_{{\alpha _{1}}}(t)T_{{\alpha _{2}}}(t). [404.1.3.4] Further generalization is possible by iterating the replacement to get \tilde{T}_{{\alpha _{1},\alpha _{2},...,\alpha _{n}}}=\tilde{T}_{{\alpha _{1},\alpha _{2},...,\alpha _{{n-1}}}}T_{{\alpha _{n}}}(t).