Editing Potts model (section)

=== Partition function and measure ===
The corresponding finite-state [[partition function (statistical mechanics)|partition function]] is given by
: <math>Z_n(V) = \sum_{s_0,\ldots,s_n \in Q} \exp(-\beta H_n(C_0[s_0,s_1,\ldots,s_n]))</math>
with ''C''<sub>0</sub> being the cylinder sets defined above. Here, β = 1/''kT'', where ''k'' is the [[Boltzmann constant]], and ''T'' is the [[temperature]]. It is very common in mathematical treatments to set β = 1, as it is easily regained by rescaling the interaction energy. This partition function is written as a function of the interaction ''V'' to emphasize that it is only a function of the interaction, and not of any specific configuration of spins.  The partition function, together with the Hamiltonian, are used to define a [[measure (mathematics)|measure]] on the Borel σ-algebra in the following way: The measure of a cylinder set, i.e. an element of the base, is given by
: <math>\mu (C_k[s_0,s_1,\ldots,s_n]) =  \frac{1}{Z_n(V)}  \exp(-\beta H_n (C_k[s_0,s_1,\ldots,s_n]))</math>

One can then extend by countable additivity to the full σ-algebra. This measure is a [[probability measure]]; it gives the likelihood of a given configuration occurring in the [[Configuration space (physics)|configuration space]] ''Q''<sup>'''Z'''</sup>. By endowing the configuration space with a probability measure built from a Hamiltonian in this way, the configuration space turns into a [[canonical ensemble]].

Most thermodynamic properties can be expressed directly in terms of the partition function. Thus, for example, the [[Helmholtz free energy]] is given by
: <math>A_n(V)=-kT \log Z_n(V)</math>

Another important related quantity is the [[topological pressure]], defined as
: <math>P(V) = \lim_{n\to\infty} \frac{1}{n} \log Z_n(V)</math>
which will show up as the logarithm of the leading eigenvalue of the [[transfer operator]] of the solution.