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<Δ⁢t<∞ for the discretization time step, and holds that “there is no essential difference between discrete-time and continuous-time systems”[24],page 51. ¹ (This is a footnote:) ¹ It is argued that one can always write t∈R as t=ϵ+n⁢Δ⁢t where ϵ=t-n⁢Δ⁢t is small and n=t/Δ⁢t is the largest integer not larger than t/Δ⁢t. As long as 0<Δ⁢t<∞ the continuous long time limit limt→∞⁡T~t corresponds to the discrete long time limit limk→∞⁡T~k⁢Δ⁢t. [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<Δ⁢t<∞ (or Δ⁢t=1). [552.2.3]The two other alternatives are Δ⁢t→0 and Δ⁢t→∞. [552.2.4]The first alternative considers the limit limΔ⁢t→0,k→∞⁡S~k⁢Δ⁢t 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 limΔ⁢t→∞,k→∞⁡S~k⁢Δ⁢t in which the discretization step diverges Δ⁢t→∞. [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

[page 553, §0]   where the last equation defines the N-fold convolution pN⁢k. [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 pN⁢k=p⁢k*…*p⁢k gives the probability density pN⁢k=ProbTN=k⁢Δ⁢t of the random variable TN=τ1+…+τN representing the sum of N independent and identically distributed random recurrence times τj with common lattice distribution p⁢k=p1⁢k. [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 p⁢k belongs to the domain of attraction of a stable density [25, 26]. [553.1.3]Then, because Δ⁢t is defined as the maximal value such that all the τi are concentrated on the arithmetic progression k⁢Δ⁢t, it follows that for a suitable choice of renormalization constants CN,DN

where h⁢x;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, h⁢x;ϖ,ζ,C,0=δ⁢x-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

which expresses the well known scaling relations for stable distributions [25, 26, 18, 20]. [553.2.4]The scaling function hϖ⁢x can be expressed explicitly as

in terms of general H-functions whose definition may be found in [28] or [18, 20]. [553.2.5]For ϖ=1 one finds

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 Λ⁢N 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→∞

[page 554, §0]   where the centering constants have been chosen conveniently as CN=-C⁢DN. [554.0.1]From this it is clear that the traditional long time limit N→∞ keeping 0<Δ⁢t<∞ finite produces limN→∞0<Δ⁢t<∞⁡k⁢Δ⁢t/D⁢N⁢Λ⁢N1/ϖ=0 for k finite, and thus limN→∞0<Δ⁢t<∞⁡pN⁢k=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 Δ⁢t to become infinite. [554.0.4]If Δ⁢t diverges such that

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→∞Δ⁢t→∞⁡k⁢pN⁢k=t⁢hϖ⁢t/D1/ϖ/D1/ϖ and thus from eq. (10) that

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

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].