As usual, let an individual state of the system be denoted by 
, let the probability of be given by 
, let averages using 
over be denoted by angle brackets, and let the context be the spin Hamiltonians of equations 6.1 and 8.1.
Distribution function:
Average energy:
Average magnetic moment:
Entropy:
From the above note that
The first relationship is the familiar dS=dQ/T from thermodynamics in different units, and indicates that the entropy is an increasing function of increasing average energy. In fact, combining equations 10.20 we have
which does not hold true for the entropy of the reduced densities, for example look at the Ising model, equation 6.30 where the derivatives of the reduced entropy per spin are explicitly found to be larger than 
at all orders. See also the discussion in chapter 2.
