Dynamics for the fluctuations
The rest of the terms are for determining the distribution of ξ. By collecting terms of O (n0) in the Taylor series expansion this equation is seen to be given by
with
and
This is a type of Fokker-Planck equation we have encountered earlier in this book.
It can be solved as discussed in Aoki (1996a, Sec. 5.13), for example. As we discuss shortly, the local equilibria of the dynamics are the zeros of the function Φ. Its derivative Φ' is negative at those local equilibria that are locally asymptotically stable, i.e., where A is positive. Note that the coefficient C is 2φb(φ) at the critical points.Equation (9.2) can be solved by the method of separation of variables. Let ∏(ξ, t) = T(t)X(ξ). Then we obtain
where θ is some constant.
To obtain a stationary solution, set θ equal to zero. Rewriting the equation for X as
In the case where X'(0) = 0, we can solve it as
We have thus shown that this stationary distribution for ξ is normally distributed with mean zero and variance C/A. Its variance is given by
With two or more locally stable equilibria, the probability mass around each of the critical points may overlap and assign positive probability to the neighboring critical points. This is one sufficient condition for fluctuations to spill over to the neighboring basins of attraction. Even if this does not happen, we show later that expected first-passage times from one basin to the neighboring ones are finite, i.e., cycles are possible. We show below that the stationary distribution for ξ is normally distributed with mean zero and variance that is a function of φ but is independent of n. This is a posteriori justification for the change of the variable we have performed.
9.5