Methods for Solving and Analyzing the Fokker–Planck Equation 237
Here, we have an element of randomness because quantity v
1 s-s
can be either positive
or negative, depending on the amplitude of small disturbances.
As was mentioned earlier, the steady-state probability distribution at R = 0 has the
form
P(v) = C exp
(
−
v
2
i
2σ
2
)
. (9.31)
Assume now R 6= 0 and R < 1. In this case, we can easily obtain that fluctuations
of components relative to their steady-state values (ev
i
= v
i
− v
i s-s
) satisfy the system
of equations
d
dt
ev
0
(t) =ev
2
2
(t) −ev
2
1
(t) −ev
0
(t) + f
0
(t),
d
dt
ev
1
(t) =ev
0
(t)ev
1
(t) − (1 − R)ev
1
(t) + f
1
(t),
d
dt
ev
2
(t) = −ev
0
(t)ev
2
(t) − (1 + R)ev
2
(t) + f
2
(t).
(9.32)
Statistical characteristics of the solution to system (9.32) can be determined using
the perturbation theory with respect to small parameter σ
2
. The second moments of
componentsev
i
will then be described by the linearized system of equations (9.32); the
second moment being determined, mean values can be obtained by averaging system
(9.32) directly. The corresponding steady-state variances of fluctuations ev
i
have the
form
D
ev
2
0
(t)
E
= σ
2
,
D
ev
2
1
(t)
E
=
σ
2
1 − R
,
D
ev
2
2
(t)
E
=
σ
2
1 + R
. (9.33)
Note that expressions (9.33) hold for R 1. With increasing R, both intensity and
temporal correlation radius of component ev
1
increase, while the intensity of compo-
nent ev
2
decreases. In this process, maximum fluctuations of quantity ev
1
occur when
the dynamic system gets over the critical regime.
Consider now the case of R > 1. The steady-state probability distribution of com-
ponent v
1
has two maxima near v
1
= ±
√
R − 1 (they correspond to the stable steady-
state positions) and a minimum v
1
= 0 that corresponds to the unstable position. This
probability distribution is formed by ensemble averaging over realizations of random
forces f
i
(t). In a single realization, the system arrives with a probability of 1/2 at one of
the stable positions corresponding to distribution maxima. In this case, averaging over
time or ensemble averaging over the force realizations that bring the system to this
state will form the probability distribution near the maximum, and we can determine
statistical characteristics of the solution using the perturbation theory in parameter σ
2
.
Assume the system arrives at the state corresponding to the stable steady-state posi-
tion v
2 s-s
= 0, v
1 s-s
=
√
R − 1. Then, fluctuations relative to this state will satisfy the