The graphs for the moments, correlation functions, and cumulants for the four spin system are given in figures 8.24-8.31. The moments show transitions at the same values of the external field that gave rise to entropy maxima. The moment graphs indicate that the antiferromegnatic region is where all of the interesting behavior occurs. The stairstep moment increase to spin alignment in the antiferromagnetic region as the external field is increased (recall that negative temperature is being used to create a negative coupling constant, and that this affects the alignment in that it makes anti-alignment with the external field more favorable; the external field has thus also effectively changed in sign.) indicates that each transition is where a subset of the energy eigenstates with increasingly aligned spins becomes dominant. The plots of the correlations and cumulants (the first three cumulant functions and correlation functions are equal by order) indicate much of the same interesting behavior as the information correlation functions. The information they convey is about average values of products of deviations, and so they have the ability to indicate this aspect of the complex structure of the antiferromagnetic region. In the ferromagnetic region the third order correlation appears to dip in the region of small external field and high temperature. This region is not brought sharply to attention by any structure in the information correlation functions in this region.
Figure 8.24: First order moment for the 4 spin case.
Figure 8.25: Second order moment for the 4 spin case.
Figure 8.26: Third order moment for the 4 spin case.
Figure 8.27: Fourth order moment for the 4 spin case.
Figure 8.28: Second order correlation for the 4 spin case.
Figure 8.29: Third order correlation for the 4 spin case.
Figure 8.30: Fourth order correlation for the 4 spin case.
Figure 8.31: Fourth order cumulant for the 4 spin case. Recall that the first three correlations and cumulants are equal by order.