Bragg-Williams microcanonical esemble

Question

In this question Bragg-Williams theory of phase transition of the forum someone was asking for Bragg-Williams aprox. and how to calculate entropy. The answer is clear and correct, the Bragg-Williams aproximation use microcanonical essemble to calculate S and then calculate F.

I wonder why you can use Microcanonical in a Spin-interaction system. Ising model is usually treat in the canonical esemble since the probability of a spin to get a negative value is not the same compared to a positive value.

Answer 1

The microcanonical ensemble is used to model a system with fixed energy, while the canonical ensemble is used to model a system with fixed temperature (and variable energy). Spin systems - like anything else - can be described by using either, depending on whether the system is allowed to exchange energy with an external thermal reservoir.

[...] the probability of a spin to get a negative value is not the same compared to a positive value.

Incidentally, if the spins are interacting with one another, then it is typically not meaningful to assign a probability to the value of a single spin, because the likelihood that a particular spin is up or down depends on the spins of its neighbors. The microcanonical and canonical ensembles assign probabilities to entire microstates of the system - that is, a specification of every spin - which can only be simplified to a probability distribution for individual spins if they don't influence one another.

Answer 2

You can always use the microcanonical ensemble. In the Ising model, this amounts to fixing some (admissible) energy $E$ and considering the uniform measure on the set of all configurations with energy $E$.

Let me take as an example the simplest situation in which the probability of up an down spins are different: the case of independent spins $\sigma_1,\dots,\sigma_N$ in a field, that is, the Ising model with Hamiltonian $H(\sigma) = -h\sum_{i=1}^N \sigma_i$. The admissible energies in this case are those in the set $\mathcal{E} = \{-Nh, (-N+2)h,\dots, (N-2)h, Nh\}$.

Given some energy $E\in\mathcal{E}$, the corresponding microcanonical measure is then simply the uniform measure on the set $$ \{\sigma\in\{-1,1\}^N\,:\, H(\sigma) = E\}. $$ Of course, in this trivial case, this set can easily be described explicitly: if $E=nh$, then a configuration has energy $E$ if and only if there are $(N-n)/2$ spins equal to $+1$ and $(N+n)/2$ spins equal to $-1$.

Of course, the general scheme remains true if you include interactions between the spins.