[635.1.1] The time evolution of almost invariant states can
be defined by the addition of random recurrence times.
[635.1.2] Let pNk
be the probability density of the sum
of N≥1 independent
and identically-distributed random
recurrence times wi≥1.
[635.1.3] Let pk from Equation (39)
be the common probability density of all wi.
[635.1.4] Then, with N≥2 and p1k=pk,
| pNk=pN-1*pk=∑m=0kpN-1mpk-m | | (41) |
is an N-fold convolution of the discrete
recurrence time density in Equation (39).
[635.1.5] The family of distributions pNk obeys
for all N≥1, and the discrete analogue of
Equation (5)
holds for all N,M≥1.
[635.1.6] Because the individual states in G are indistinguishable
within the given accuracy η, but may evolve very
differently in time, it is natural to define the duration
of time needed for the first recurrence (a single time step) as an average
over recurrence times.
[635.1.7] If a macroscopic time evolution with a rescaled
time exists, then one has to rescale the sums
WN and the iterations
| S-N=S-N-1S-1=∑k=1∞pNkT-k | | (45) |
in the limit N→∞ with suitable norming constants DN≥0.
Theorem 3.
[635.1.8] Let pNk be the probability density of WN
specified above in (41).
[635.1.9] If the distributions of WN/DN converge
to a limit as N→∞ for suitable norming constants DN≥0,
then there exist constants D≥0 and 0<α≤1,
such that
| limN→∞supkDNpNk-τD1/αhαkτDND1/α=0 | | (46) |
[page 636, §0]
where:
| α=sup0<β<1:∑k=1∞kβpk<∞ | | (47) |
if ∑k=1∞kpk diverges, while
if ∑k=1∞kpk converges.
[636.0.1] For α=1, the function hαx is h1x=δx-1.
[636.0.2] For 0<α<1, the function hαx=0 for x≤0 and
| hαx=1x∑j=0∞-1jx-αjj!Γ-αj | | (49) |
for x>0.
Proof.
[636.1.1] The existence of a limiting distribution for WN/DN>0
is known to be equivalent to the stability of the limit [18].
[636.1.2] If the limit distribution is nondegenerate,
this implies that
the rescaling constants DN have the form
where ΛN is a slowly varying function [19],
defined by the requirement that
holds for all b>0.
[636.1.3] That the number α obeys Equation (47)
is proven in [18] (p. 179).
[636.1.4] It is bounded
as 0<α≤1, because the rescaled random
variables WN/DN>0 are positive.
[636.2.1] To prove Equation (46), note that
the characteristic function of WN is
the N-th power
| eiξWN=pξN=∑keiξypNk | | (52) |
because the characteristic functions
pξ=eiξwj
of wj are identical for all j=1,…,N.
[636.2.2] Inverse Fourier transformation gives:
| pNk=12π∫-ππe-iξkpξNdξ=τ2πDND1/α∫-πDND1/απDND1/αe-iξxpξτDND1/αNdξ | | (53) |
where:
and ξ was substituted with ξτ/DND1/α.
[636.2.3] Let hαξ denote the characteristic function of
hαx, so that
| hαx=12π∫-∞∞e-ixξhαξdξ | | (55) |
holds.
[page 637, §1]
[637.1.1] Following [20],
the difference ΔNk in (46) can be
decomposed and bounded from above as
| ΔNk | =|DNpN(k)-τD1/αhα(kτDND1/α)|=|DNpN(k)-τD1/αhα(x)| | |
| =τ2πD1/α∫-πDND1/απDND1/αe-iξxpξτDND1/αNdξ-∫-∞∞e-iξxhαξdξ | |
| =τ2πD1/α|∫ξ<Be-iξx[p(ξτDND1/α)N-hα(ξ)]dξ+∫B≤ξ<ηDND1/αe-iξx[p(ξτDND1/α)N-hα(ξ)]dξ | |
| +∫η≤ξDND1/α<πe-iξx[p(ξτDND1/α)N-hα(ξ)]dξ-∫ξ≥πDND1/αe-iξxhα(ξ)dξ| | |
| ≤τ2πD1/α(∫ξ<B|p(ξτDND1/α)N-hα(ξ)|dξ+∫B≤ξ<ηDND1/α|p(ξτDND1/α)|Ndξ | |
| +∫η≤ξDND1/α<π|p(ξτDND1/α)|Ndξ+∫ξ≥Bhα(ξ)dξ) | | (56) |
with constants B,η to be specified below.
[637.1.2] The terms involving hαξ from the second and third
integral have been absorbed in the fourth integral.
[637.1.3] The four integrals are now discussed further individually.
[637.2.1] The first integral converges uniformly to zero for
N→∞, because
pk belongs to the domain of attraction
of a stable law with index α, as already noted above.
[637.3.1] To estimate the second integral, note that the characteristic
function pξ belongs to the domain of attraction
for index α if and only if it behaves for ξ→0
as [20]
where c>0 and Λx
is a slowly varying function at infinity obeying
[637.3.2] By the representation
theorem for slowly varying functions ([21] p. 12),
there exist functions dy and εy,
such that the function Λy can be represented as
| Λy=dyexp-∫byεuudu | | (59) |
[page 638, §0]
for some b>0 where dy is measurable and
dy→d∈0,∞, as well as εu→0
hold for y→∞.
[638.0.1] As a consequence
| ΛλyΛy=dλydyexp-∫yλyεuudu | | (60) |
so that with λ=ξ-1 and y=DN
| ΛDN/ξΛDN=ξo11+o1 | | (61) |
is obtained for N→∞.
[638.0.2] Therefore, there exists for any γ<α a positive number
cγ independent of N, such that
| pNξ=pξDNN=exp-cNDNαΛDNΛDNΛDNξξα≤exp-cγξγ | | (62) |
for sufficiently large N.
[638.0.3] If N is sufficiently large,
it is then possible to choose an η>0
(and find c~γ), such that
| ∫B≤ξ<ηDND1/αpξτDND1/αNdξ≤∫B≤ξ<ηDND1/αexp-c~α2ξα2dξ≤∫ξ≥Bexp-c~α2ξα2dξ | | (63) |
and this converges to zero for B→∞.
[638.1.1] The third integral is estimated by noting that
pξ<1 for 0<ξ<2π/τ.
[638.1.2] Hence, there is a positive constant c>0, such that
for η≤ξ≤π.
[638.1.3] Consequently, with Equation (50),
| ∫η≤ξDND1/α<πpξτDND1/αNdξ≤2πe-cNNΛND1/α | | (65) |
converges to zero as N→∞.
[638.2.1] Finally, the fourth integral converges to zero, because
the characteristic function hαξ is integrable on R.
[638.2.2] In summary, all four terms in Equation (56)
vanish for N→∞, and Equation (46) holds.
∎
[638.3.1] Equation (46) implies
| pNk≈τDND1/αhαkτDND1/α | | (66) |
for sufficiently large N and all τ.
[638.3.2] Inserting this into Equation (45) gives
| S-N=∑k=1∞pNkT-k≈∑k=1∞τDND1/αhαkτDND1/αT-k=∑k=1∞hαkτDND1/αT-kk-k-1τDND1/α | | (67) |
[page 639, §0]
[639.0.1] For α=1, the average return time
τ∑kkpk<∞
is proportional to the discretization τ.
[639.0.2] In the case 0<α<1, the average
time τ∑kkpk=∞ for return into
the set G in a single step diverges.
[639.0.3] This suggests an infinite rescaling of time as τ→∞
for 0<α<1.
[639.0.4] This rescaling of time combined with
N→∞ was called the ultra-long-time limit in [22].
[639.0.5] In the ultra-long-time limit N→∞,τ→∞ with:
| limτ→∞N→∞DND1/ατ=limτ→∞N→∞NΛND1/ατ=a | | (68) |
one finds from Equation (67) the result
| limτ→∞,N→∞NΛND1/α/τ=aS-N≈∑k=1∞hαkaT-kak-k-1a≈∫0∞hαxT-xadx | | (69) |
for sufficiently large N and τ.
[639.0.6] The limit gives rise to
a family of one-parameter semigroups Tαa
(with family index α and parameter a)
of ultra-long-time evolution operators
| limτ→∞,N→∞NΛND1/α/τ=aS-N=Tα-a=∫0∞hαxT-xadx | | (70) |
which are convolutions instead of translations.
[639.0.7] Note that
a≥0 because
DN≥0 and D≥0.
[639.0.8] The rescaled age evolutions Tα-a
are called fractional time evolutions, because
their infinitesimal generators are
fractional time derivatives [22, 23].
[639.1.1] The result shows that a proper mathematical
formulation of local stationarity requires a generalization of the
left-hand side in Equation (1), because
Equation (1) assumes implicitly a translation along the orbit.
[639.1.2] In general, the integration of infinitesimal
system changes leads to convolutions instead
of just translations along the orbit [22, 23].
[639.1.3] Of course, translations are a special case of
convolutions, to which they reduce in the
case when the parameter α approaches unity.
[639.1.4] For α→1-, one finds
| h1x=limα→1-hαx=δx-1 | | (71) |
and therefore
| T1-a=∫0∞δx-1T-xadx=T-a | | (72) |
is a right translation.
[639.1.5] Here, a≥0 is an age or duration.
[639.1.6] This shows that also the special case of
induced right translations does not give
a group, but only a semigroup.
