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2 Fractional time evolutions

[400.2.1.1] What does it mean to replace an ordinary time derivative with a fractional derivative ? [400.2.1.2] Are fractional time derivatives the infinitesimal generators of translations or other symmetry transformations, and, if yes, what is their nature ? [400.2.1.3] Which fractional derivative should be used ?

[400.2.2.1] These questions have been generally neglected by all workers in the field, and were only recently addressed and answered in Refs. [21, 22, 4, 5, 3]. [400.2.2.2] It was found that generalized fractional time evolutions T_{\alpha}, whose infinitesimal generators are fractional time derivatives of order \alpha, arise very generally in the transition between microscopic and macroscopic time scales. [400.2.2.3] The fractional time evolution T_{\alpha}(t) for duration t is defined through its action on an observable f(t_{0}) depending on the time instants t_{0} by [21, 22, 4, 5, 3]

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

where t\geq 0 and 0<\alpha\leq 1. [400.2.2.4] The kernel function h_{\alpha}(x) is the one sided stable probability density with stable index \alpha [21, 22, 4, 5, 3]. [400.2.2.5] Its Mellin transform is known to be [23]

\mathscr M\{ h_{\alpha}(x)\}(t_{0})=\frac{1}{\alpha}\frac{\Gamma((1-s)/\alpha)}{\Gamma(1-s)}. (2)

[400.2.2.6] This allows to identify its density function as [24, 22, 4, 5, 25]

h_{\alpha}(x)=\frac{1}{\alpha x}H_{{11}}^{{10}}\left(\frac{1}{x}\left|\begin{array}[]{l}{(0,1)}\\
{(0,1/\alpha)}\end{array}\right.\right) (3)

in terms of H-functions [26, 27]. [400.2.2.7] Its well known Laplace transform reads

\mathscr L\{ h_{\alpha}(x)\}(u)=e^{{-u^{\alpha}}}. (4)

[400.2.2.8] The operators T_{\alpha}(t) form a semi-group and obey the basic semi-group relation [page 401, §0]

T_{\alpha}(t_{1})T_{\alpha}(t_{2})=T_{\alpha}(t_{1}+t_{2}). (5)

[401.1.0.1] The infinitesimal generator A_{\alpha} of the fractional semi-group T_{\alpha}

A_{\alpha}f(t)=-(D^{{\alpha}}f)(t)=-\frac{1}{\Gamma(-\alpha)}\int _{0}^{\infty}\frac{f(t-s)-f(t)}{s^{{\alpha+1}}}\;\mathrm{d}s (6)

is the fractional Marchaud-Hadamard derivative [28]. [401.1.0.2] For \alpha=1 the fractional semi-group T_{1}(t) becomes the semi-group T(t)f(t_{0})=f(t-t_{0}) of simple translations. [401.1.0.3] Because of this and because of the properties (1) and (5) the fractional semi-group T_{\alpha}(t) will also be called “fractional translation” for short.

[401.1.1.1] The fractional time evolution/translation T_{\alpha}(t) seems to have been first introduced into physics in connection with the discovery of a new class of phase transitions [21]. [401.1.1.2] It was later derived for dynamical systems from ergodic theory in [22, 4, 5]. [401.1.1.3] Based on these results it was argued that fractional time evolutions and fractional dynamics actually exist in nature. [401.1.1.4] Recently the physical basis for formula (1) was generalized further using the idea of coarse graining [3]. [401.1.1.5] Formula (1) was previously known in pure mathematics where it has close connections with the theory of semi-groups and subordination [29, 30]. [401.1.1.6] It did not find direct applications in physics until the present author used it as the foundation for the theory of fractional time evolutions in physics. [401.1.1.7] Formula (1) was recently rediscovered in physics in the more restricted context of fractional diffusion [31].