and for ℓ = 1 this simplifies to Compare this with (66). We thus have a = − 2 e − 2 $^{K}$. Now y and z are related as y = 2 w − a where w is defined by (68). If we set K = 2 / 15 then a = − 1 . 531857, and choosing z = − 3 . 537 indeed gives us y = 1 . 01002. To actually solve z as a function of K seems harder though. However, numerical experimentation suggests that y and z for small K behave as where c$_{1}$ ≈ 6 . 164 and c$_{2}$ ≈ − 10 . 33. Strangely, when the p, q -distribution fit the magnetisation distribution so well one might think that the free energy would be well approximated by (139). This is not so. The p, q -approximation differs clearly from the asymptotic free energy, given by log (2 cosh K ). For the 2-dimensional lattices we can rely on exact data only for up to L = 16 and they were computed according to the method in $^{14}$. We have sampled data for L = 32 , 64 , 128 , 256 , 512, collected with the methods described in 15 and $^{16}$. These meth- ods gave us the energy distribution and then it is just a matter of combining this with the distribution of magnetisations for each given energy as described in $^{12}$. Figure 10 shows an example of some distributions for the 128 × 128-lattice together with their fitted p, q -binomial distributions. The fit is fairly good, but hardly excellent near K ∗ . However, as the figure shows, at K = 0 . 4388 (i.e. for L = 128) the fit is practically spot on. For the lattices we have studied there is always one such temperature where the p, q -distribution fit particularly well. This point is located between K ∗ and K$_{c}$ and is very close to, but not exactly equal to, the point where the susceptibility is at its maximum. Of course, for high temperatures (small K ) and low temperatures (high K ) the fit is typ- ically very good but in the high-temperature region the measured y and z are unfortunately extremely sensitive to noise. As we get closer to the critical region where the distribution becomes bimodal this problem goes away, even though the sampled distributions are more B. 2D-lattices 38 y ( K ) ≈ $_{c$_{1}$}$√$_{K}$ + c$_{2}$ K (148) z ( K ) ≈ − $_{c$_{1}$}$√$_{K}$ + c$_{2}$ K (149) R ( n, n/ 2 , 1) = exp ( − 1 / 2 σ $^{2}$) = exp ( − 2 e − 2 $^{K}$/n ) ∼ 1 − 2 e − 2 K n (147)