Correlation Function and Generalized Master Equation of Arbitrary Age Page: 4
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ALLEGRINI et aL. PHYSICAL REVIEW E 71, 066109 (2005)
result using arguments based on trajectories rather than den-
sities, thereby affording an independent construction of the
exact expression for the t,-old correlation function All this
can be thought of as a compelling demonstration of the cor-
rectness of Eq (11), extending the Onsager principle to con-
ditions of arbitrary age
B. Derivation of the exact expression proposed by Godr&che
and Luck: The probability perspective
To establish that the proposed approach yields the exact
expression of Godreche and Luck [14], let us express the
t-old correlation function through the probability vector p
(pi,p2) for the dichotomous variable 5= 1 to have either
positive (state 1) or negative (state 2) values We have that
(2 pi (O)p (t1,t=0)+p(O)2 (t2,t=0)
pl(O)p2(tl1,t= O) p(O)pl(tl2,t=0), (12)
where pj(t k, t= 0) is the conditional probability that the varin-
able 4 is in the state j at time t, given that at time t 0 it was
in the state k This means that p(tl k, t= 0) is obtained letting
those trajectories evolve that at time t 0 had in the state k
For a straightforward evaluation of pj(tlk,t 0), we use the
GME formalism, adapted to the t,-old system, and we take
into account the initial condition p,(0 k, t 0)= 8, Accord-
ing to the GME the components of the conditional probabil-
ity vector are determined byp,(uIk,0)
[u+( 4t(u)],+ ,(U) ( ),K+1 + 84,K 1)
u[u+ 2(b (u)](15)
Using Eq (15) we can Laplace transform (12) to obtain
ae)(u) = (0) u+ (u) + (0)
u(u+ 24(t(u)) u(u+ 24t(u))+4o(u)
u(u+ 24 (u))%,(u)
p2(o) U
u(u+ 24ft(u))We note that the probability is normalized, pl(0)+p2(0)= 1
Thus, it follows thatu(u+ 24( (u))
X[P1(0) + P2(0)]S2tu)
u(u+ 24 (u))ju+2,(u)
confirming the correctness of the definition introduced in Eq
(10) Substituting (9) into (17) we obtainBt(.)
I 1 + 2 at(u)
l 1+ (u) 2t(u)dt' it(t-t' ,p,(t'k,t=O),
aO r1with fta(u) given by Eq (9) and the elements of K given by
Eq (3)
By Laplace transforming (13) and doing some algebra, we
obtain
2
,(uk,0)=! [ul+ t,(u)K],pt(OIk,O) (14)
Defining the matrixu+ t,(u)
J= [uI+4(u)K] 1= u[u+2t(u)]
a+ 2(u)
u[u+ 2( "(u)]t,()
u[u+ 2t4(u)]
u[u+ 24~t(u)]and using the mitial condition p,(O0 k, t= 0)= s,, we obtain
for the Laplace transform of the conditional probability vec-
tor the following expression1 2 beai(u)
u 1+(u)
(18)which coincides with the results of Godreche and Luck [14]
Furthermore, the ratio of the differences in probability is
determined in Laplace space byPl(u) -P(u)
p1(0) pZ(0)S1 pJ) p2()0)
P() -P2(0)1
u+ 2,()(19)
As pointed out earlier, Eq (19) means that one can extend
the Onsager principle from the infinitely aged systems, for
which Onsager originally defined it, to systems of any age
In the latter case the relaxation is proportional to the t-old
correlation function, not to the infinitely old, or equilibrium,
correlation function In summary, we discovered an Onsager
principle of arbitrary age, at least m the special case of the
dichotomous variables considered m this paper
C. Derivation of the exact expression proposed by Godr&che
and Luck: The trajectory perspective
It is possible to again derive the exact result of Eq (18)
from a different perspective, which will allow us, in Sec IV,
to propose an analytic expression for the t,-old correlation066109-4
p(tlk, t= 0)
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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/m1/4/: accessed April 2, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT College of Arts and Sciences.