Information Technology Reference
In-Depth Information
Figure 7.2.
The equilibrium positions as a function of the coupling parameter
K
,
redrawn from [
15
] with
permission.
By inserting the definition of the transition rates given by (
7.16
)into(
7.18
) we obtain
the mean-field equation for the web of two-state nodes,
d
dt
=−
)
=−
∂
U
()
∂
,
2
g
cosh
(
K
)
−
2
g
sinh
(
K
(7.19)
which corresponds to the dynamics of a particle in a double-well potential introduced
in Chapter 3.
Equation (
7.19
) describes the overdamped motion of a particle, whose “position” is
, and the values of its minima depend only on
the coupling constant
K
. It is straightforward to show that there is a critical value of the
coupling parameter given by
K
within the symmetric potential
U
()
=
K
c
=
1
,
such that (1) if
K
≤
K
c
the potential has
only one minimum located at
=
0; or (2) if
K
>
K
c
the potential is symmetric and
has two minima located at
±
min
separated by a barrier with the maximum centered at
=
The potential has the same form as the quartic potential shown in Figure
3.4
and
the height of the potential barrier is a monotonic function of the coupling parameter
K
.
The time evolution of the dynamical variable
0
.
is determined by the extrema of
the potential and consequently two kinds of dynamical evolution are possible: (1) if
K
≤
K
c
, the dynamical variable
(
t
)
will, after a brief transient, settle to an asymp-
totic value
(
∞
)
=
0 independently of the initial condition
(
0
)
;
(2) if
K
>
K
c
,the
dynamical variable
(
t
)
will, after a transient, reach the asymptotic value
(
∞
)
=
min
=
0 for an initial condition
(
0
)>
0
,(
∞
)
=−
min
=
0 for an initial condi-
tion
(
0
)<
0 and finally
(
t
)
=
0 for all time given an initial condition
(
0
)
=
0. In
Figure
7.2
we compare the minima
±
min
and the numerical evaluation of
(
∞
)
for
various values of the coupling constant
K
.
The location of the potential minima as a function of the coupling parameter
K
is
given in Figure
7.2
, which shows that a phase transition occurs at
K
=
K
c
=
1. For
a single two-state node the condition
(
∞
)
=±
min
=
p
1
−
p
2
=
0 corresponds to
the statistical preference of the particle to be in either the state
r
2.
This is a consequence of the transition rates being different if the coupling parameter
is greater than the critical value. By inserting the mean-field value of the probabilities
(
7.15
) into the expression for the transition rates (
7.16
) and allowing the dynamical
variable to reach its asymptotic value we obtain
=
1 or the state
r
=
ge
−
K
(
∞
)
=
ge
K
(
∞
)
.
g
12
=
g
21
=
(7.20)
