Once the eigenstates of the Hamiltonian are found and written in
a form where 
is easily found then the distributions for seeing values
of 
are readily generated. First, the energy of each of the energy eigenvectors is found. From these energies the probabilities of the eigenvectors at equilibrium are given by the Boltzmann distribution 
. The distribution of the vectors of values is then generated. From this distibution the correlation functions and information correlation functions desired may be computed as a function of temperature and coupling strength.
The distribution of measured spin values is simply found from the
representation of the energy eigenvectors in the 
basis. Given an eigenvector in the energy eigenbasis, the probability of measuring the eigenvalues 
is the square of the amplitude of the component vector 
, where the 
values for 
are ignored. Then sum this probability times the probability that the energy eigenvector occurs (Boltzmann distribution) over the energy eigenvectors to find the overall probability of the measured values. This is the same thing as tracing over the reduced density matrix for spins 
times the operator 
, i.e.
Evaluating the information correlation functions (see section 3.12) is trivial because the information correlation functions are simply a sum involving functions of the probabilities of the observed states, which is what is generated in equation 8.12 above.
Evaluating the moments, and therefore the correlations and cumulants (see section 8.12) is straightforward. In evaluating the average of a product of spins 
each term involving an even number of 
values contributes a positive quantity. Every other term contributes a negative quantity. The magnitude of each term is 
times the probability of the state. The moments are given by the trace
