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8 Fractional Time Evolution

[page 214, §1]
[214.1.1] The induced time evolution is obtained from GT by iteration. [214.1.2] According to its definition in eq. (16) the induced measure preserving transformation GT acts as a convolution in time,

 G⁢T⁢ϱ=ϱ*p (17)

where ϱ is a mixed state on G,S. [214.1.3] Iterating N times gives

 G⁢TN⁢ϱ=G⁢TN-1⁢ϱ*p=ϱ*p⁢…*p︸N⁢ factors=ϱ*pN (18)

where pNk=pk*pk is the probability density of the sum

 TN=τ1+…+τN (19)

of N independent and identically with pk distributed random recurrence times τi. [214.1.4] Then the long time limit N for induced measure preserving transformations on subsets of small measure is generally governed by well known local limit theorems for convolutions [42, 43, 44, 45]. [214.1.5] Application to the case at hand yields the following fundamental theorem of fractional dynamics [1, 21]

Theorem 8.1.

Assume that τ>0 is maximal in the sense that there is no larger τ for which all recurrence times lie in τN. [214.1.6] Then the following conditions are equivalent:

1. [214.1.7] Either k=1kpk< or there exists a number 0<γ<1 such that

 γ=sup⁡0<β<1:∑k=1∞kβ⁢p⁢k<∞. (20)
2. [214.1.8] There exist constants DN0,D0 and 0<α1 such that

 limN→∞⁡supk⁡DNτ⁢p⁢k-1D1/α⁢hα⁢k⁢τDN⁢D1/α=0 (21)

where α=1, if k=1kpk<, and α=γ otherwise. [214.1.9] The function hαx vanishes for x0, and is

 hα⁢x=1x⁢∑j=0∞-1j⁢x-α⁢jj!⁢Γ⁢-α⁢j. (22)

for x>0.

[page 215, §0]    [215.0.1] If the limit exists, and is nondegenerate, i.e. D0, then the rescaling constants DN have the form

 DN=N⁢Λ⁢N1/α (23)

where ΛN is a slowly varying function [46], i.e.

 limx→∞⁡Λ⁢b⁢xΛ⁢x=1 (24)

for all b>0.

[215.1.1] The theorem shows that

 pN⁢k≈τDN⁢D1/α⁢hα⁢k⁢τDN⁢D1/α (25)

holds for sufficiently large N. [215.1.2] The asymptotic behaviour of the iterated induced measure preserving transformation GTN for N allows to remove the discretization, and to find the induced continuous time evolution on subsets GΓ. [215.1.3] First, the definition eq. (15) is extended from the arithmetic progression s˙-τN to t˙s˙ by linear interpolation. [215.1.4] Let ϱ~t˙ denote the extended measure defined for t˙s˙. [215.1.5] Using eq. (11) and setting

 t=DN⁢D1/α (26)

the summation in eq. (18) can be approximated for sufficently large N by an integral. [215.1.6] Then GTNϱ~s˙GTαtϱ~s˙, where

 G⁢Tαt⁢ϱ~⁢s˙=∫0∞ϱ~⁢s˙-t˙⁢hα⁢t˙t⁢d⁢t˙t (27)

is the induced continuous time evolution.[215.1.7]  GTαt is also called fractional time evolution. [215.1.8] Laplace tranformation shows that GTαt fulfills eq. (5). [215.1.9] It is an example of subordination of semigroups [47, 33, 7, 48]. [215.1.10] Indeed

 G⁢Tαt=1t⁢∫0∞Tt˙⁢hα⁢t˙t⁢d⁢t˙ (28)

where Tt˙ denotes right translations on the interpolated measure. [215.1.11] Because DN0 and D0, eq. (26) implies t0. [215.1.12] As remarked in the introduction, the induced time evolution is in general not a translation (group or semigroup), but a convolution semigroup. [215.1.13] The fundamental classification parameter

 α=α⁢T,G,τ (29)

[page 216, §0]    depends not only on the dynamical rule T(,t˙) and the subset G, but also on the discretization time step τ, i.e. on the time scale of interest.