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of the perturbation. In general this solution is not exact, but under certain conditions
this prediction turns out to be correct for sufficiently weak perturbations. We take up
the general theory of this approach in later chapters.
3.2.3
Double-well potential and stochastic resonance
The next step in increasing modeling complexity is to make the potential of order higher
than quadratic. A nonlinear oscillator potential coupled to the environment is given by
a dynamical equation of the form
d
2
Q
dt
2
dQ
dt
−
∂
U
(
Q
)
=−
λ
Q
+
ξ(
t
),
(3.92)
∂
so, in addition to the potential, the particle experiences fluctuations and dissipation due
to the coupling to the environment. Of most interest for present and future discussion is
the double-well potential
Q
2
a
2
2
A
4
U
(
Q
)
=
−
,
(3.93)
which has minima at
Q
0. In the Smoluchosky approxi-
mation the inertial force, that is the second-derivative term in (
3.92
), is negligibly small
so that (
3.92
) can be replaced with
dQ
=±
a
and a maximum at
Q
=
dt
=−
∂(
Q
)
Q
+
η(
t
),
(3.94)
∂
where
η(
t
)
=
ξ(
t
)/λ
and the scaled potential is
(
Q
)
=
U
(
Q
)/λ
so that the dynamics
are determined by
Q
2
a
2
Q
dQ
dt
=−
A
λ
−
+
η(
t
),
(3.95)
where the fluctuations and dissipation terms model the coupling to the environment.
The Smoluchosky approximation is widely used in stochastic physics. One of the
most popular applications is in the
synergetics
of Herman Haken [
21
]. An essen-
tial concept in synergetics is the order parameter, which was originally introduced
to describe phase transitions in thermodynamics in the Ginzberg-Landau theory. The
order-parameter concept was generalized by Haken to the
enslaving principle
, which
states that the dynamics of fast-relaxing (stable) modes are completely determined
by a handful of slow dynamics of the “order parameter” (unstable modes) [
22
]. The
order dynamics can be interpreted as the amplitude of the unstable modes determining
the macroscopic pattern. The enslaving principle is an extension of the Smoluchosky
approximation from the physics of condensed matter to complex networks, including
the applications made for understanding the workings of the brain [
23
,
24
].
The problem of a thermally activated escape over a barrier was studied by Kramers in
his seminal paper [
28
] and studies are continually revealing insights into such complex
phenomena; see for example [
33
]. The theory provides a touchstone against which to
assess more elaborate analyses, since it provides one of the few analytic results avail-
able. Consequently, we record here the evaluation of the rate of escape over a potential
