Dynamics for the aggregate variable
We mention here only the dynamics for the conditional mean of the fraction of agents with choice 1 in situations where P(n, t) is expected to have a well- defined peak of order N, and fluctuations or variation of order √N.
The differential equations for the fluctuations are introduced later in connection with the search model. To capture this feature, we change variables from n(t) to
where φ(t) is the mean of the fraction, and ξ is a random variable to represent fluctuation about the mean. The idea of this scaling is to make φ and ξ about 0(1). Let
with n defined above. We have
where the partial derivative with respect to time is taken keeping n fixed, i.e., we have
To simplify our derivation of approximate dynamics, regroup the right-hand side of the master equation by the lead and lag operators
and
as
Then, note that the change from n to n + 1 entails change of ξ by a factor
Thus, for any smoothly differentiable function a(n), 
Analogously, we write
The expressions for the transition rates are also expanded into Taylor series.
To balance the orders in N, we change the time scale by
Then, we equate the highest-order term in N, which is
, with the terms on
the right of the same order to obtain the differential equation for φ, which was introduced at the beginning of this section in changing the variable from n to ξ:
This is the dynamic equation for the aggregated variable, the fraction.[6]
This aggregate equation is thus obtained by equating the highest-order terms from each side of the master equation. It is more comprehensive and more informative than a set of equations for the moments, which can be easily derived. See Aoki (1995) and Aoki (1996a, p. 136) for more detail. If the right-hand side is identically zero, we see that φ(τ) = φ(0) for positive τ, that is, any small deviation in φ(0) does not decay to zero. In this case we need to use a different time scaling than the one used above in expanding the master equation. This leads to what is known as the diffusion approximation. See Aoki 11996a, Sec. 5.4).
As an example, consider transition rates given by
and
where n is now the number of agents making choice 1, N is the total number of agents in the model, and where η's are some function of n/N. At the boundaries, we set rN = 0 and l0 = 0 because the model is closed with no entry and no exit. Then, we derive as the right-hand side of the aggregate dynamics
There are two differences between the transition rates in the previous closed model and those of this section, one minor and the other major.
The function h(n) is introduced to represent some scale effects. This is a minor point, and we set it to 1 without loss of generality, since it can be absorbed into the units of time. The η's are a major difference. They represent externalities and the perceived advantages of alternative choices as functions of the fraction k/n, i.e., the state variable of the model.In the next section, we show that the zeros of this function G (φ) satisfy the same equation as the critical points of the potential of the stationary distribution 
that is, the critical points of the potential and the zeros of G are the same. See Aoki (1996a, pp. 118, 140) for further points on potentials. To pick out the critical points that correspond to local minima, we sign the derivative of G. Calculating its derivative
we obtain the condition for the local asymptotic stability of the critical points as
5.3