that is, an infinite string of spins. In the above example, the function V just picked out two spins out of the infinite string: the values s0 and s1. In general, the function V may depend on some or all of the spins; currently, only those that depend on a finite number are exactly solvable. Define the function Hn : QZ → R as This function can be seen to consist of two parts: the self-energy of a configuration [s0, s1, ..., sn] of spins, plus the interaction energy of this set and all the other spins in the lattice. The n → ∞ limit of this function is the Hamiltonian of the system; for finite n, these are sometimes called the finite state Hamiltonians. Partition function and measure The corresponding finite-state partition function is given by with C0 being the cylinder sets defined above. Here, β = 1/kT, where k is Boltzmann's 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 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 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 QZ. 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 Another important related quantity is the topological pressure, defined as which will show up as the logarithm of the leading eigenvalue of the transfer operator of the solution. Free field solution The simplest model is the model where there is no interaction at all, and so V = c and Hn = c (with c constant and independent of any spin configuration). The partition function becomes If all states are allowed, that is, the