Figure 3.

We show how the probabilities associated with the numerically determined probability density change when the spacing of discrete frequencies, ε, is reduced. We considered the piecewise constant probability distribution at a discrete time corresponding to t = 100 generations. To make the numerical calculation of all probabilities as comparable as possible, we set the ratio α of Equation 5, which characterizes the numerical scheme, to have the value α = 500. We then determined the time step, for a given value of ε, from τ = 2ε²α and the time index from n = 100/τ. Thus different points of the figure are associated with different ε and hence different τ and n, but the values of t and α are held fixed. The probabilities associated with the last two bins on the right in Figure 1, namely bin K − 1 and bin K, are $p_{K-1}\left(\epsilon \right)=\epsilon f_{K-1}^n$ and $p_K\left(\epsilon \right)=\left(\epsilon /2\right)f_K^n$, respectively. We observe in Figure 3 that when ε approaches 0, the probability p_K−1(ε), associated with bin K − 1, converges to a small number that, within numerical error, may be taken as zero. However the probability p_K(ε), associated with the end bin, approaches an appreciable nonzero value. This is precisely what we would expect if the end bin contains a spike (a Dirac delta function) whose entire weight is located at x = 1 and which always contributes to the probability of the bin, as long as its width is positive. By contrast, the probability associated with the adjacent bin (bin K − 1) has the behavior we would expect of a smooth probability distribution, i.e., one that does not contain a spike. The lines through the data points in the figure result from fitting a quadratic function of ε to p_K−1(ε) and a linear function of ε to p_K(ε).