We may also find the thermodynamic limit of the k-spin system. The algebra is omitted here because it's similar to the algebra in the 
case.
Note that taking 
here gives . Graphs of the entropy per spin for orders 1-4 including both the ferromagnetic Q>0 and 
and the antiferromagnetic Q<0 and (Q>0 and 
) appear in figures 6.1- 6.4. Note the sharp difference in the antiferromagnetic case between the first and second order entropies, and the marked similarities between the second and higher order entropies. Note that the logarithms are base e, and that 
appears as b in the axis label.
Figure 6.1: First order entropy per spin of the Ising system. Entropy of one spin.
Figure 6.2: Second order entropy per spin of the Ising system. Note the difference between the first and second order entropies, and the similarities of the second-fourth order entropies. The bump on the antiferromagnetic phase side occurs as the external field is increased, and indicates a transition between the 
states, and the 
state.
Figure 6.3: Third order entropy per spin of the Ising system.
Figure 6.4: Fourth order entropy per spin of the Ising system.