Renewal aging and linear response Page: 2
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with the inverse power law form:
(T) (u- 1)T )' (1)
with p > 1 and
T> 1. (2)
The condition of Eq. (2) is essential to adopt, whenever convenient, the earlier mentioned continuum time treatment.
The probability that no event occurs until time t is
(t) - dt'/(t') = t+T "(3)
This function is called survival probability.
When p > 2, the mean value of these waiting times is given by
< >= 2) (4)
The condition ,u = 2, where this mean value diverges is the border between the region ,u > 2, compatible with the
existence of an infinitely aged condition , and the region ,u < 2, which is characterized by perennial aging and
ergodicity breakdown . In this paper we focus our attention on the condition of perennial aging ,u < 2.
For a detailed discussion of renewal aging we refer the reader to the recent literature on this subject, for instance,
Refs. [13, 14]. The proper quantitative treatment of aging requires preparation, namely, that the time origin t 0
corresponds to an event occurrence. Aging means that the waiting time distribution 4(T) and the corresponding
survival probability W(T) depend on the time at which the observation process begins. We define with the symbol
/(t, t') the probability of meeting the first event at time t when the observation begins at time t' < t and the system
is prepared at time t = 0. W(t, t') denotes the corresponding survival probability. The functions b(t, t') and TI(t, t')
refer to a condition where t = ti, namely, a time at which an event occurs, and t' is a time at which no event occurs
and observation begins. Of course, the Poisson conditions 4b(t, t') 4= (t - t') and TI(t, t') = T(t - t') are violated
[13, 14] but in the specific case where t' = ti_1, namely, when observation begins with an event occurrence. More
precisely, the functions 4(T) and W(T) refer to the condition T = ti - ti-1, with t2, denoting, according to the earlier
definitions, times at which events occur. To discuss the response of renewal non-Poisson systems to perturbations we
study their behavior in the presence of a time dependent perturbation, a condition that forces us to introduce also
the function b(t t'). This is the conditional probability that at time t an event occurs, given the condition that the
earlier event occurs at time t'. In the limiting case of extremely weak perturbation it is expected that
(t)t') = 0(t - t'). (5)
In fact, in the unperturbed case, as earlier pointed out, the waiting time distribution is obtained by recording the
distance between an event and the next, insofar as any time can be selected as a time origin, if it corresponds to
an event occurrence. Renewal aging does not have anything to do with the system being driven by physical rules
changing with time, and it is only a consequence of making observation and preparation at different times. This
condition, and the equality of Eq. (5) with it, is violated by external perturbation. Let us note furthermore that in
the literature the symbols T and b are usually adopted to denote a probability and a probability density, respectively.
Due to the adoption of a discrete time representation in this article both symbols denote probabilities, insofar as, for
instance, <(T)dT = (7), as a consequence of the fact that for the integration time step we have dT = 1.
The problem of the response of sub-diffusion to external perturbation in an aging condition has been recently
addressed by Sokolov, Blumen and Klafter  and by Barkai and Cheng . These authors have adopted the
same phenomenological model as Bertin and Bouchaud . In this model the external perturbation, which does
not affect the sojourn time duration, influences the choice between either jumping to the right or to the left, at the
end of a sojourn: In the unperturbed case there is no bias, this choice being done with a fair coin, and the effect
of perturbation is to turn this fair coin into an unfair coin, producing a bias at the moment of the choice between
the left and the right nearest-neighbor site. The phenomenological method will be extended to the velocity picture.
However, in addition to adopting the velocity picture, and thus the physical condition of super-diffusion rather than
sub-diffusion, we plan to go beyond the phenomenological model. For this reason, we shall also study the case when
the perturbation-induced bias is generated through the influence that the external perturbation exerts on the waiting
time duration. Although this is in principle a hard problem, we find an analytical solution in the limiting case of weak
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Allegrini, Paolo; Ascolani, Gianluca; Bologna, Mauro & Grigolini, Paolo. Renewal aging and linear response, paper, February 6, 2008; (https://digital.library.unt.edu/ark:/67531/metadc174702/m1/2/: accessed March 18, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.