The derivatives with respect to 
of the reduced entropy are not ordered. This can be seen by rewriting the expression of equation 10.23 as
Consider the second term of equation 10.26. Is it always the same sign? The answer to this is no, as may be seen more clearly by expanding the energy difference as
The first term of equation 10.27 substituted into the second term of equation 10.26 is immediately positive for the same reason that the derivative of the full entropy is negative. However, the second term of equation 10.27 substituted into the second term of equation 10.26 is guaranteed neither positive nor negative. Nor does its magnitude appear easily bounded. See figures 10.1 and 10.2 for an example of a region where the derivative of the first order entropy changes sign while the derivative of the second order entropy does not. Note that the same may hold true in general for derivatives of the reduced entropies with respect to the other parameters, as figures 8.4 and 8.17 show clearly. Thus, although the reduced entropies have an ordering, the derivatives of them do not.
Figure 10.1: First order entropy of Ising system in cross section in the antiferromagnetic region. Note the change in the sign of the derivative with respect to . Considering a neighboring pair of sites, this occurs as the temperature is lowered as the states 
, 
, 
, 
, which yield equal probabilities of 
and 
at a single site, get replaced by the smaller set of states , , , which yield a greater probability for than at a single site, and these in turn finally replaced by and alone, which again yield equal probabilities of and at a single site.
Figure 10.2: Second order entropy of Ising system in cross section in the antiferromagnetic region. Note that the derivative of the entropy with respect to maintains the same sign.