Substituting for the various quantities with R=0 in the expression for
and simplifying we find
where 
and 
are defined as
In terms of the R=0 value of 
(see equation 6.23) we have
From equation 6.30 we see that the entropy per spin decreases as the order increases, consistent with the reduced entropy per degree of freedom theorem of chapter 2. From the form of equation 6.30 it is implied that the change in entropy with changes in the average energy (or any parameter) decreases in magnitude as the order increases. Thus it is important to observe the high-order structure indicated by the high order entropy in order to understand the creation of that structure as the average energy changes. In fact, since 
(see chapter 10, equation 10.22), note that
Note the extremely simple k-dependence of the kth order entropy .
Note that the quantities and obey
so that they may be interpreted as probabilities, and referring to equation 6.5 we see that and are the probabilities that any two neighboring spins are the same and different respectively. Note that 
.
Note that all derivatives of with respect to system parameters or variables have the same functional
dependence on system parameters up to the factor 1-1/k.