Correlation Function and Generalized Master Equation of Arbitrary Age Page: 11
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CORRELATION FUNCTION AND GENERALIZED MASTER PHYSICAL REVIEW E 71, 066109 (2005)
symmetric part of the distribution free to evolve If the dis-
tribution is at equilibrium, the symmetric part corresponds to
the equilibrium distribution and the integral of the left por-
tion of the antisymmetric part, without back injection, re-
gresses to zero as the corresponding equilibrium correlation
function For any other condition, the integral of the left
portion of the antisymmetric part regresses to zero with an
analytical expression depending on the time at which obser-
vation begins The regression continues as a function of that
specific initial condition while the symmetric part keeps
moving towards equilibrium independently of the population
difference This explains why the regression to equilibrium
depends on the imtial condition, of any age, with no further
dependence on the bath dynamics that keeps drifting towards
equilibrium This also explains why an emission or absorp-
tion spectrum [11] is not stationary and changes with time
The resonant radiation establishes a connection between the
antisymmetric and the symmetric parts of the distribution,
thereby updating observation to the changing bath condi-
tions
It is worth ending this paper with some further remarks
about these theoretical problems We have built up a GME of
arbitrary age, using an empirical approach Is it possible to
derive the same GME by using a Llouville-like approach? In
principle, we should use the Llouville-lihke picture of Sec III,
to derive, via contraction on the bath variables, the same
GME, of arbitrary age, as that of Sec II D However, it is
evident that this effort, even if we were successful, would be
of limited help, for practical purposes Suppose, for instance,
that we have to study the response of the system to an exter-
nal, time dependent, perturbation Would the GME of arbi-
trary, but fixed, age, be useful for this purpose? It is evident
that it would not In fact, the external perturbation at times
different from the age of the system would produce effects
departing from the more realistic approach resting on per-
turbing trajectories In the specific case of the absorption
spectrum of blinking quantum dots [11] the authors, in fact,
adopted this trajectory perspective to make a theoretical pre-
diction that is incompatible with the perturbation of a GME
of fixed ageThere already exists in the literature at least the discussion
of one case [19] that seems to be a natural consequence of
this property Sokolov, Blumen, and Klafter [19] derived an
exact density equation to describe a subdiffusion process
This equation corresponds to brand new initial conditions
This means a condition where t,= 0 Then, these authors per-
turbed this equation with a time-dependent field, and found
that the theoretical result conflicts with the behavior of the
CTRW under the influence of the same perturbation This
apparent contradiction arises because the time dependent
perturbation corresponds to additional observations, taking
place at different time values, none of them coinciding with
the observation time, but the perturbation at t 0 This is true,
whatever the observation time is, either t= 0, as in Ref [19],
or co> t,>0, a condition requirmg the GME of this paper
Regardless of the observation time that we assign to the
GME, it is impossible to make the GME prediction identical
to the CTRW prediction, if we require the perturbation to
remain external to the system The concept of perturbation
itself turns out to be inadequate to study non-Poisson pro-
cesses, regardless of its intensity Thus the results of the
present paper, in addition to shedding light onto the implica-
tions of Ref [19], imply a violation of linear response theory
The only possible way to make the density compatible with
the trajectory picture is to make the external perturbation
become a part of the system under study This means that we
have to build up a totally new, field-dependent, GME, along
the lines of Ref [11] This sets a limit on the applicability of
the GME of arbitrary age found in this paper However, this
result seems to support the conclusion that the trajectory-
density conflict, revealed by Bologna, Grigohml and West
[13] might be a consequence of the aging properties emerg-
ing from non-Poisson renewal process
ACKNOWLEDGMENTS
GA and PG thankfully acknowledge ARO for financial
support through Grant DAAD19-02-1-0037 PG acknowl-
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Allegrini, Paolo; Aquino, Gerardo; Grigolini, Paolo; Palatella, Luigi; Rosa, Angelo & West, Bruce J. Correlation Function and Generalized Master Equation of Arbitrary Age, article, June 10, 2005; [College Park, Maryland]. (https://digital.library.unt.edu/ark:/67531/metadc40401/m1/11/: accessed April 24, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.