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# 3 Averaged Induced Dynamics in the Ultralong Time Limit

[552.1.1]The induced transformations and 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 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 as where is small and is the largest integer not larger than . As long as the continuous long time limit corresponds to the discrete long time limit . [552.1.3]Obviously, this equivalence between discrete and continuous time breaks down for induced dynamics because the continuous flow of time within is interrupted by time periods of fluctuating length during which the trajectory wanders outside . [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 (or ). [552.2.3]The two other alternatives are and . [552.2.4]The first alternative considers the limit 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 in which the discretization step diverges . [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 acts as a convolution operator in time

 (9)

[552.3.2]Applying the transformation times yields

 (10)

[page 553, §0]   where the last equation defines the -fold convolution . [553.0.1]If exists this defines also in the long time limit.

[553.1.1]To determine whether a limiting density exists, note that the -fold convolution gives the probability density Prob of the random variable representing the sum of independent and identically distributed random recurrence times with common lattice distribution . [553.1.2]A necessary and sufficient condition for the existence of a limiting density for suitably renormalized recurrence times is that the discrete lattice probability density belongs to the domain of attraction of a stable density [25, 26]. [553.1.3]Then, because is defined as the maximal value such that all the are concentrated on the arithmetic progression , it follows that for a suitable choice of renormalization constants

 (11)

where is a limiting stable density whose parameters obey , , , and [25, 26, 27]. [553.1.4]If then the limiting distribution is degenerate, for all values of .

[553.2.1]The positivity of the recurrence times for all implies that the renormalized recurrence times are bounded below, and this gives rise to the constraint for on the possible limiting distributions. [553.2.2]The limiting stable distributions compatible with this constraint are given by those with parameters and . [553.2.3]For the limiting densities may be abbreviated as

 (12)

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

 (13)

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

 (14)

the Dirac distribution concentrated at as the limiting density. [553.2.6]If the limit exists and is nondegenerate, i.e , the renormalization constants must have the form

 (15)

where is a slowly varying function [26], defined by the condition that

 (16)

for all .

[553.3.1]Using equations (11) and (12) one has for

 (17)

[page 554, §0]   where the centering constants have been chosen conveniently as . [554.0.1]From this it is clear that the traditional long time limit keeping finite produces for finite, and thus , unless . [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 to become infinite. [554.0.4]If diverges such that

 (18)

exists, then this defines a renormalized ultralong continuous time, . [554.0.5]In this case contrary to the conventional limit. [554.0.6]It follows that and thus from eq. (10) that

 (19)

where the ultralong time parameter was identified as

 (20)

[554.0.7]The identification of is justified for two reasons. [554.0.8]On the one hand for all , where is the expectation with respect to the limiting distribution, and are two independent random recurrence times. [554.0.9]This shows that has dimensions of time. [554.0.10]Secondly for it follows from (14) that

 (21)

which again identifies 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 resp. [554.0.12] 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].