3 Averaged Induced Dynamics in the Ultralong Time Limit
[552.1.1]The induced transformations S~G and SG were defined
for discrete time, and it is of interest to remove the discretization
to obtain the induced dynamics in continuous time.
[552.1.2]The conventional view on discrete vs. continuous time
in ergodic theory assumes 0<Δt<∞
for the discretization time step, and holds that
“there is no essential difference between
discrete-time and continuous-time systems”[24],page 51.
[552.1.3]Obviously, this equivalence between discrete and
continuous time breaks down for induced dynamics
because the continuous flow of time within G is
interrupted by time periods of fluctuating length
during which the trajectory wanders outside G.
[552.1.4]These interruptions produce a discontinuous
(fluctuating) flow of time.
[552.2.1]There are three possibilities for removing the discretization
using a long time limit. [552.2.2]Only one of these employs the
conventional assumption 0<Δt<∞ (or Δt=1). [552.2.3]The two
other alternatives are Δt→0 and Δt→∞. [552.2.4]The first alternative considers the limit
limΔt→0,k→∞S~kΔt in which the discretization
step becomes small. [552.2.5]This possibility may be called the
short-long-time limit or continuous time limit,
and it was discussed in [21]. [552.2.6]The second alternative
is to consider the limit limΔt→∞,k→∞S~kΔt
in which the discretization step diverges Δt→∞. [552.2.7]This
will be considered in this paper, and it is called the
long-long-time limit or the ultralong-time limit. [552.2.8]These limits are analogous to the ensemble limit
[18, 19, 20, 21].
[552.3.1]According to its definition (8) the induced time
transformation SG acts as a convolution operator in time
[552.3.2]Applying the transformation N times yields
SGNϱB=SGN-1ϱB*p=ϱB*p*…*p︸Nfactors=ϱB*pN |
| (10) |
[page 553, §0] where the last equation defines the N-fold convolution pNk. [553.0.1]If p∞=limN→∞pN exists this defines also SGN
in the N→∞ long time limit.
[553.1.1]To determine whether a limiting density p∞ exists, note that
the N-fold convolution pNk=pk*…*pk gives the probability
density pNk=ProbTN=kΔt of the random variable
TN=τ1+…+τN
representing the sum of N independent and identically
distributed random recurrence times τj with common
lattice distribution pk=p1k. [553.1.2]A necessary and
sufficient condition for the existence of a limiting
density p∞ for suitably renormalized recurrence
times is that the discrete lattice probability density
pk belongs to the domain of attraction of a stable
density [25, 26]. [553.1.3]Then, because Δt is defined as
the maximal value such that all the τi are concentrated
on the arithmetic progression kΔt, it follows that for a
suitable choice of renormalization constants CN,DN
limN→∞supkDNΔtpNk-hkΔt-CNDN;ϖ,ζ,C,D=0 |
| (11) |
where hx;ϖ,ζ,C,D is a limiting stable density
whose parameters obey 0<ϖ≤2, -1≤ζ≤1,
-∞<C<∞, and D≥0 [25, 26, 27]. [553.1.4]If D=0 then the
limiting distribution is degenerate, hx;ϖ,ζ,C,0=δx-C
for all values of ϖ,ζ.
[553.2.1]The positivity of the recurrence times τi≥0 for all
i∈N implies that the renormalized recurrence times TN
are bounded below, and this gives rise to the constraint
P∞t=0 for t≤C
on the possible limiting distributions. [553.2.2]The limiting stable
distributions compatible with this constraint are given by
those with parameters 0<ϖ≤1 and ζ=-1. [553.2.3]For 0<ϖ<1 the limiting densities may be abbreviated as
hx;ϖ,-1,C,D=1D1/ϖhϖt-CD1/ϖ |
| (12) |
which expresses the well known scaling relations for stable
distributions [25, 26, 18, 20]. [553.2.4]The scaling function hϖx can be expressed explicitly as
hϖ(x)=1xϖH1110(1x|0,10,1/ϖ) |
| (13) |
in terms of general H-functions whose definition may be found
in [28] or [18, 20]. [553.2.5]For ϖ=1 one finds
h1x=limϖ→1-hϖx=δx-1 |
| (14) |
the Dirac distribution concentrated at x=1 as the limiting
density. [553.2.6]If the limit exists and is nondegenerate,
i.e D≠0, the renormalization constants DN must have the form
where ΛN is a slowly varying function [26], defined by
the condition that
for all b>0.
[553.3.1]Using equations (11) and (12) one has for N→∞
pNk≈ΔtDNhkΔt-CNDN;ϖ,-1,C,D=ΔtDND1/ϖhϖkΔtDND1/ϖ |
| (17) |
[page 554, §0] where the centering constants have been chosen conveniently as CN=-CDN. [554.0.1]From this it is clear that the traditional long time limit N→∞
keeping 0<Δt<∞ finite produces
limN→∞0<Δt<∞kΔt/DNΛN1/ϖ=0
for k finite, and thus limN→∞0<Δt<∞pNk=0,
unless D=0. [554.0.2]Therefore the conventional long time limit produces
a degenerate limiting distribution if it exists. [554.0.3]The ultralong
time limit on the other hand allows Δt to become infinite. [554.0.4]If Δt diverges such that
limN→∞Δt→∞kΔtDN=t |
| (18) |
exists, then this defines a renormalized ultralong continuous time,
0<t<∞. [554.0.5]In this case D>0 contrary to the conventional limit. [554.0.6]It follows that
limN→∞Δt→∞kpNk=thϖt/D1/ϖ/D1/ϖ
and thus from eq. (10) that
Sϖt*ϱB,t0* | = | ∫0∞ϱB,t0*-thϖtt*dtt* |
| (19) |
| = | 1t*∫0∞TtϱB,t0*hϖt/t*ds |
|
where the ultralong time parameter t* was identified as
[554.0.7]The identification of t* is justified for two reasons. [554.0.8]On the one hand
D∝τ-τ′σϖ/σ
for all σ<ϖ, where … is the expectation
with respect to the limiting distribution, and τ,τ′
are two independent random recurrence times. [554.0.9]This shows that
D1/ϖ has dimensions of time. [554.0.10]Secondly for ϖ=1 it
follows from (14) that
S1t*ϱB,t0*=∫-∞∞ϱB,t0*-tδtt*-1dtt*=ϱB,t0*-t*=Tt*ϱB,t0* |
| (21) |
which again identifies t*=D1/ϖ as an ultralong time
parameter. [554.0.11]Note that the results (19) and (21)
imply macroscopic (=ultralong time) irreversibility by virtue
of (20) even if the underlying time evolution
T~t resp. [554.0.12]Tt was reversible. [554.0.13]Perhaps this could provide
new insight into the longstanding irreversibility paradox. [554.0.14]The fundamental convolution semigroup (19) was first
obtained in [18, 19] and [21].