Ising Model and Partition FUnction

[A Dhingra]

Hi all

This is probably a naïve question to ask, but I am puzzled by it and need an answer.
The first time I encountered the term 'partition function' that was in context of Boltzmann distribution. But the same formulas of manipulating a partition function ( to obtain free energy, temperature etc.) are used in 1D or 2D (or more) Ising model. It is quite odd that Ising model is about magnetic interactions of spin 1/2 particles, which are fermions, then why do we use partition function with the Boltzmann weight... Shouldn't there be something in the partition function or elsewhere that tells us which distribution we are talking about?
Further, I want to know how is this idea of partition function derived (or obtained) for a Boltzmann distribution, and other than simplifying the problem what is the relevance of this partition function in any distribution?

thanks for any help...

 

[DrClaude]

First, in the Ising model the particles are fixed, distinguishable and in different states, whatever the spin. There is therefore no particular care to be taken that they are fermions.

Second, the Boltzmann factor for a state $s$, $\exp[-\beta E(s)]$, where $E(s)]$ is the energy of the state, is a "universal" function relating the energy of a state and the probability of it being populated at a temperature $T$ (in the canonical ensemble). It has a very wide application.

The partition function is basically the normalization factor that is needed to convert the Boltzmann factor to a probability, and is the sum of Boltzmann factors over all states.