Power-series expansion

As above we expanded the transition rate in inverse powers of N, we may expand the master equations in N^-1 to solve it by retaining terms only up to O (N^-1).³ In some models, we anticipate that the probability density, the time evolution of which is governed by the master equation, will show a well-defined peak at some X of order N, with spread (standard deviation) of order √N, if the initial condition is

In such cases, we change the variable by introducing two variables φ and ξ, both of order one, and set (recall that x(t) = X(t)/N)

Later we show that φ is the mean of the distribution when this change of the variable is applicable, i.e., φ(t) keeps track of the mean of x (t), and the spread about the mean is expressed by a random variable ξ (t).

We next show that the terms generated in the power-series expansion of the master equation separate into two parts. The first part, which is the larger in magnitude, is an ordinary differential equation for φ. This is interpreted to

^[1] When these terms are zero, we may want to retain terms of order N^-2. Then we have diffusion­equation approximations to the master equation. See Aoki (1996a, Sec. 5.14) for diffusion approximations.

be the macroeconomic or aggregate equation. The remaining part is a partial differential equation for ξ with coefficients that are functions of φ, the first term of which is known as the Fokker-Planck equation.

We rewrite the probability density for ξ as

by substitutingIn rewriting the master equation for ∏ we

must take the partial derivative with respect to time keeping x(t) fixed, i.e., we must impose the relation

and we obtain

We also note that we need to rescale time by

Otherwise, the random variable ξ will not be O (N⁰), contrary to our assumption, and the power-series expansion will not be valid. Butwealsoassume f (N) = N in this section. In general it is the case that τ = t. We use τ from now on to accommodate this more general scaling function.

The master equation in the new notation is given by

⁴ We need not be precise about the initial condition, since an expression of O(N^-1) or O (N^-1/2) can be shifted between the two terms without any consequence. Put differently, the location of the peak of the distribution can’t be defined more precisely than the width of the distribution, which is of order N^1/2. See Section 9.6 for an illustration.

⁵ Following the common convention that the parameters of the density are not carried as arguments in the density expression, we do not explicitly show φ when the substitution is made.

3.3