Absorption and Emission in the Non-Poissonian Case Page: 3
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VOLUME 93, NUMBER 5
PHYSICAL REVIEW LETTERS
Ref. [9] studied the aged condition f(t, oo) with 2 < v <
3, and found that the corresponding spectrum has two
sharp peaks. Klafter and Zumofen, in Ref. [10], proved
that the same case with f(t, to 0) /= (t) yields Levy
diffusion, thereby implying an exponential characteristic
function, and consequently a Lorentzian spectrum. It is
remarkable that our theoretical perspective establishes a
connection between the prediction of Zumofen and
Klafter, valid at short times, and that of Ref. [9], valid
at large times. Remarkably, we evaluate numerically also
the spectrum time evolution in the case v < 2, where the
process of aging keeps going on forever without ever
reaching the stationary condition, which does not exist,
in this case (see Fig. 2).
Let us show now how to reproduce these numerical
results with a proper theoretical treatment. Let us consider
as an example a single trajectory starting a time t' with
s +W and ending at time t with the same positive
Ivalue after n switches of the variable e between its two
values _W. In this case the integrand of (5), has the
following form:e[a+(t- t')+a (t2 tl)+...+ +(t- t)]
(8)
where a = i(6 + W) and n can only be even.
To determine the contribution to ( t(t)) of Eq. (6) stem-
ming from this kind of trajectories, we have to average
the term (8) on the set of all possible sequences of the
variable e(t) running from t' to t with n switches, to sum
over n and then to carry out the integration on t'. To this
purpose, as earlier stated, besides the probability density
i(7) of having an interval T between two generic con-
secutive switching events, it is necessary to use f(r, t'),
this being the conditional probability density that, fixed a
time t', the first next switching event of the variable S(t)
occurs at time t' + 7. The result of this procedure turns
the contribution to (6) intof t dt F(t - t', t')e a+(t-t') + ft dt1f(t1
n=L 1where, in addition to the crucial probability density
f(r, t'), we have usedThis is given by
This is given byS(T) = 1- o(T')dT',
(10)
which is the conventional probability that no switch
occurs for a generic interval of time 7, andF(r, t') = 1 - f(', t')dr',
which is the corresponding aging property, dependir
f(r, t'), and indicating therefore the conditional prob
ity that, fixed t', no switch occurs between t = t' anc
t' + 7. The overall factor of 1/2 of the contribution
a consequence of the fact that at time t' the fluctu
e(t), supposed to be positive, can get with the
probability the negative value. Let us address now
problem of finding an exact analytical expression fo
crucial property f(r, t'). The exact expression for f(
is
f(r, t') = t dr'G(t'- 7')/I(T + 7'),
where
G(t) 8(t) + #(t) + n=2 fdtz(t)
X dt20(t2 - tl)" dtn-l (t- tn-
1 n-2(11)
lg on
abil-
It=
(9) is
ation
samen=0
1
1 - i(u)(14)
where #(u) denotes the Laplace transform of fi(t). Thus,
the Laplace transform of (12) with respect to t' reads
f(, u') = eu (u) - e-u (y)dy,
1 - (u') Loj
(15)thereby yielding
f(, t') = -1[f(T, u')].(16)
the With this expression the prescription necessary to evalu-
r the ate the contribution to ( (t)) of Eq. (6) of the trajectories
(r, t) beginning in the "light on" state at t' and ending in the
same state at t, is completed.
As a last step, we calculate the Laplace transform of
(12) (9), and of the equivalent expressions for all the other
possible conditions of motion from t' to t, namely, with
the noise e(t) moving from + W to - W, from - W to + W
and, finally, from -W to -W, as well. This procedure
yields, as a final result, the Laplace transform of R(t),
denoted by 1^(u), which is proved to have the following
analytical expression:1)"
(13)
It is straightforward to find the Laplace transform of G(t).
050601-3R(u) k[(,i(t) + ,*(t))]
k2
-[A+ (u) + A _(u) + C(u)].
2(17)
050601-3
week ending
30 JULY 2004t2)ea +(t3 t2) . . .
(9)
t', t')ea+(tl-t') ftdt20(t2
1lt )ea (t2-tl) tdt3 (t3
t2X dt2n (t2n - t2n-l)ea (t2n-t2n-1) l(t - t2njea+(t-t2n) ,
2n -1
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Aquino, Gerardo; Palatella, Luigi & Grigolini, Paolo. Absorption and Emission in the Non-Poissonian Case, article, July 28, 2004; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc67641/m1/3/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.