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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 Tα, whose infinitesimal generators are fractional time derivatives of order α, arise very generally in the transition between microscopic and macroscopic time scales. [400.2.2.3] The fractional time evolution Tαt for duration t is defined through its action on an observable ft0 depending on the time instants t0 by [21, 22, 4, 5, 3]

 Tα⁢t⁢f⁢t0=∫0∞f⁢t0-s⁢hα⁢st⁢d⁢st (1)

where t0 and 0<α1. [400.2.2.4] The kernel function hαx is the one sided stable probability density with stable index α [21, 22, 4, 5, 3]. [400.2.2.5] Its Mellin transform is known to be [23]

 M⁢hα⁢x⁢t0=1α⁢Γ⁢1-s/αΓ⁢1-s. (2)

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

 hα(x)=1α⁢xH1110(1x|0,10,1/α) (3)

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

 L⁢hα⁢x⁢u=e-uα. (4)

[400.2.2.8] The operators Tαt form a semi-group and obey the basic semi-group relation [page 401, §0]

 Tα⁢t1⁢Tα⁢t2=Tα⁢t1+t2. (5)

[401.1.0.1] The infinitesimal generator Aα of the fractional semi-group Tα

 Aα⁢f⁢t=-Dα⁢f⁢t=-1Γ⁢-α⁢∫0∞f⁢t-s-f⁢tsα+1⁢d⁢s (6)

is the fractional Marchaud-Hadamard derivative [28]. [401.1.0.2] For α=1 the fractional semi-group T1t becomes the semi-group Ttft0=ft-t0 of simple translations. [401.1.0.3] Because of this and because of the properties (1) and (5) the fractional semi-group Tαt will also be called “fractional translation” for short.

[401.1.1.1] The fractional time evolution/translation Tαt 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].