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8 Fractional Time Evolution

[page 214, §1]   
[214.1.1] The induced time evolution is obtained from G⁢T by iteration. [214.1.2] According to its definition in eq. (16) the induced measure preserving transformation G⁢T 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 pN⁢k=p⁢k⁢…*p⁢k is the probability density of the sum

TN=τ1+…+τN(19)

of N independent and identically with p⁢k 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 τ⁢N. [214.1.6] Then the following conditions are equivalent:

  1. [214.1.7] Either ∑k=1∞k⁢p⁢k<∞ 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 DN≥0,D≥0 and 0<α≤1 such that

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

    where α=1, if ∑k=1∞k⁢p⁢k<∞, and α=γ otherwise. [214.1.9] The function hα⁢x vanishes for x≤0, 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. D≠0, then the rescaling constants DN have the form

DN=N⁢Λ⁢N1/α(23)

where Λ⁢N 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 G⁢TN for N→∞ allows to remove the discretization, and to find the induced continuous time evolution on subsets G⊂Γ. [215.1.3] First, the definition eq. (15) is extended from the arithmetic progression s˙-τ⁢N 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 G⁢TN⁢ϱ~⁢s˙≈G⁢Tαt⁢ϱ~⁢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]  G⁢Tαt is also called fractional time evolution. [215.1.8] Laplace tranformation shows that G⁢Tαt 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 DN≥0 and D≥0, eq. (26) implies t≥0. [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.