Information Technology Reference
In-Depth Information
Going back to the difference variable and inserting the perturbed matrix elements into
(
7.61
) gives the perturbed GME
t
(
)
d
t
dt
(
t
)(
t
),
=
εξ
p
(
)
(
)
−
−
t
R
t
t
(7.64)
dt
0
where we see that the perturbation appears as a coefficient of the previously introduced
event-generation function
t
∞
dt
(
t
)
=
R
(
t
)
=
1
ψ
n
(
t
).
(7.65)
0
n
=
The inhomogeneous term in (
7.64
) gives a time-dependent coefficient to the pertur-
bation strength that depends on the number of events that occur in the time interval
(
.
It is useful to note that the Poisson condition (
7.58
) inserted into the last term in
(
7.65
) generates
0
,
t
)
R
(
t
)
=
g
.
(7.66)
Thus, in the ordinary Poisson case the number of events generated per unit time is
constant, in spite of adopting the preparation prescription at time
t
=
0, which is a form
of the out-of-equilibrium condition. That is not the case when 2
namely the
case in which the network
S
generates crucial events. Here we see that the condition
μ>
<μ<
3
,
2 is less dramatic than the condition
μ<
2. In fact, when
μ>
2, the network
tends towards the Poisson condition
1
τ
,
R
(
∞
)
=
(7.67)
but it takes an infinitely long time to do so. Note that the constant in (
7.67
) can be
associated with the constant in (
7.66
) and consequently the statistics become Poisson
asymptotically. It is reasonable to imagine that the response to perturbation of the net-
work during this infinitely extended regression to equilibrium departs from the ordinary
linear response prescription.
The condition
μ<
2 implies an even stronger departure from the conventional con-
dition. In fact, in this case the network lives in a perennial out-of-equilibrium condition.
In the case
μ>
2 it is possible to prepare the network in a stationary condition. In the
case
2 the stationary condition is not possible.
Let us again consider the memoryless structure of the classical master equation by
assuming that the memory kernel is given by the delta function (
7.59
) so that the
perturbed GME (
7.64
) reduces to
μ<
d
(
t
)
=
ε
g
ξ
p
(
t
)
−
g
(
t
),
(7.68)
dt
which has the exact solution
g
t
0
dt
e
−
g
(
t
−
t
)
ξ
p
(
e
−
gt
t
).
(
t
)
=
(
0
)
+
ε
(7.69)
