Information Technology Reference
In-Depth Information
Note that Kramers' theory affords insight into how to express the pre-factor
R
, which in
the case of the double-well potential can be shown to be the product of the potential fre-
quency at the bottom of the well and that at the top of the barrier divided by the friction
is determined by
the Einstein relation to be the thermodynamic temperature
k
B
T
. Of course, this is only
the case in discussing a physical web and is not true in general, as we shall show. Here
we expand the perturbation piece of the exponential and replace (
7.86
) with
. The diffusion coefficient
D
divided by the dissipation parameter
g
±
(
)
=
g
0
(
∓
ε
(ω
)),
t
1
cos
t
(7.87)
where we limit ourselves to stressing that
ε
1,
Q
0
D
ka
ε
=
,
(7.88)
and the unperturbed coupling parameter is
R
exp
Q
0
D
g
0
=
−
.
(7.89)
We are naturally led to replace (
7.60
) with
d
dt
p
(
)
=
(
)
(
),
t
g
0
K
t
p
t
(7.90)
where the time-dependent matrix
K
(
t
)
now has the form
−[
1
−
ε
cos
(ω
t
)
]
/
2
+[
1
+
ε
cos
(ω
t
)
]
/
2
K
(
t
)
=
.
(7.91)
+[
1
−
ε
cos
(ω
t
)
]
/
2
−[
1
+
ε
cos
(ω
t
)
]
/
2
Using the perturbed form of the coupling matrix (
7.91
)in(
7.90
) yields the equation of
motion for the difference variable
d
dt
(
t
)
=−
g
0
(
t
)
+
g
0
ε
cos
(ω
t
),
(7.92)
where
p
1
+
p
2
=
1 is used in the second term. The exact solution to this equation is
g
0
t
0
(
t
)
=
exp
(
−
g
0
t
)(
0
)
+
ε
ds
exp
(
−
g
0
(
t
−
s
))
cos
(ω
s
).
(7.93)
Let us assume that initially the web is in equilibrium so the probability of being in
either of the wells is the same and consequently
(
0
)
=
0. We define the variable
ξ
s
(
t
)
,
which has the value 1 when the network
S
is in the right well and the value
1 when
the network
S
is in the left well. Thus, we write for the average web response (
7.93
)in
the familiar LRT form
−
t
0
χ(
t
)ξ
p
(
t
)
dt
,
ξ
s
(
)
=
ε
t
t
,
(7.94)
where the response function is
t
)
=
t
)
]
χ(
t
,
g
0
exp
[−
g
0
(
t
−
(7.95)
