I am not sure whether it is some well-known named model in statistical physics. I could not find it in any standard text-book that I know of.
Let there be $N$ identical classical particles occupying a square lattice of $2N$ sites with at most one particle per site. Let the alternate sites be names $A$ and $B$. Let $p$ be the fraction of particles on $A$-type sites and assume that for a fixed value of $p$, all configurations are equally likely. (mean field approximation?)
- Then how does one calculate the entropy as a function of $p$? I guess there is some simplification that will happen if one works in the limit of $N$ being very large. (thermodynamic limit?)
Let there be a repulsive interaction energy of $E_0$ whenever neighbouring $A$ and $B$ type sites are occupied.
- In that case the claim seems to be that the average total energy $E(p)$ is given by $4NE_0p(1-p)$. (..I am not sure whether here any large $N$ limit has been taken..but I guess this too is some asymptotic answer..)
I would be glad if someone can help calculate/derive the above two bulleted questions. I would like to know if there is some general framework or class of models from which this comes.
Apparently the interesting thing about this model is that it has a second order phase transition. How does one prove that? Or is it somehow supposed to be obvious?
Will some single unique value of $p$ get picked out in the high and low temperature limit? How does one determine that?
For the last/above two of the questions one would obviously need to assume that the system is at a thermal equilibrium at a temperature $T$. Then I guess the value of $p$ gets determined by minimizing the ``free energy" $=F(p) = E(p) - TS(p)$. But to be able to do this calculation one would probably first need to find the answer to the first two questions in the large $N$/ thermodynamic limit.