# Estimating Partition functions

I have a finite state ensemble with an energy functional (you can think of it as an ferromagnetic Ising model if you like), and I need very careful estimates of the partition function. What methods are available to me to get reasonable estimates (in reasonable temperature regimes) of $Z(\beta)$?

• Why do you need the numerical value of the partition function? It's just a normalization factor -- there's a number of ways at getting at any sort of observable without explicit reference to the value of Z.
– wsc
Jan 18, 2011 at 23:56

Good question! The reponse really depends on what you 'need': if you're sampling a small system and you're lucky enough to actually be able to count (or estimate...) the number of states $\mathcal{N}(E)$ for a given energy $E$, then you can explicitly work out $Z$. Such algorithms exist for the Ising model, but in the generic case you're not this lucky.
There is the possibility of 'naively sampling' the integral/sum you're doing: you can use any Monte Carlo algorithm (especially for Ising-like systems the literature is very rich) to generate a sequence of states and you use this representative sample to calculate $Z$ for any range of temperatures. (It's important though that you do not change $\beta$ during an MC run, since this tends to break detailed balance.)