Two Dimensional Ising Model and Hamiltonian.

Question

In the Hamiltonian formalism of classical mechanics is well known that Hamiltonian function has several properties with physical interpretations. When I speak of Hamiltonian'm talking about the `` natural'' Hamiltonian of n particles with index $ i = 1, \dots, n $ that are in a region $ \Lambda \subset \mathbb{R}^3$, $$ H_\Lambda(q,p)=K_\Lambda(p)+U_\Lambda(q) $$ where $ \quad q=(q_1\dots,q_n)$, $p=(p_1,\dots,p_n)$. Here $q_i \in \mathbb{R}^3$ is the position of $i$-th particle, $p_i\in\mathbb{R}^3$ the time of the $i$-th particle, $m_i$ is the mass of each particle, $ U_\Lambda(q)=\sum_{i = 1}^n m_i\cdot q_i\cdot g $ where $g$ the gravitational acceleration and $ K_\Lambda(p)=\frac{1}{2}\sum_{i = 1}^n \frac{p_i^2}{m_i} $ the total kinetic energy of the n particles. In general it is considered $ m_i = 1 $ for all $i=1,\dots,n$.

Now the thermodynamic formalism of statistical mechanics the Ising model Hamiltonian two dimensions (see for example, Aizenman in his famous paper ``Translation Invariance and Instability of Phase Coexistence In the Two Dimensional Ising System") is given by $$ H_{\Lambda}^{\omega}(\sigma)=-\frac{1}{2}\sum_{\substack{i,j\in\Lambda \\ |i-j|=1}} \sigma_i \sigma_j - \sum_{\substack {i \in \Lambda \\ j\in\mathbb {Z} ^ 2 \\ | i-j | = 1}} \sigma_i \sigma_j $$ and when the magnetic field $ h = (h_i) _ {i \in \Lambda} $ if I remember correctly we have $$ H_ {\Lambda}^{\omega}(\sigma)=-\frac{1}{2}\sum_{\substack{i,j\in \Lambda \\ |i-j|=1}} \sigma_i \sigma_j-\sum_{\substack{i\in\Lambda \\ j\in \mathbb{Z}^2\\|i-j|=1}} \sigma_i\sigma_j-\sum_{i \in \Lambda} h_i \sigma_i $$ The analogies that can pass in and $ H_\Lambda$ for $H_\Lambda^\omega$ are as follows. Do I replace $\mathbb{R}^3$ by $\mathbb{Z}^2$, do I replace positions in $\mathbb{R}^3$ by sites in $\mathbb{Z}^2$, and do I replace variables $(q,p)$ random variables by $\sigma_i$ with $i\in \Lambda\subset\mathbb{Z}^2$. And at this point that my analogies end.

Question. In the Hamiltonian of the two dimencional Ising model who is the corresponding analogous to kinetic energy $ K (p) $ and potential energy $U(q) $?

Answer 1

You can find an heuristic argument connecting this two Hamiltonians formalism in page 2 of Ruelle's book Statistical Mechanics - Rigorous Results. I guess that Yakov Sinai discuss the same thing in more details in his book: Topics in Ergodic Theory.

Anyway, I think you will enjoy read the post (linked below) too, where people are discussing what are the "justifying foundations of statistical mechanics without appealing to the ergodic hypothesis":

https://physics.stackexchange.com/questions/27402/what-are-the-justifying-foundations-of-statistical-mechanics-without-appealing-t

Answer 2

You should consider that you are working with a discrete model and so, a connection like the one you are asking for can be meaningful if you can take the continuum limit in some way. There are fundamentally two ways to work out a problem like this. The first one is a mean field approximation. You can find some details here. The other approach is obtained starting from a scalar field theory that has a Hamiltonian

$$H=\int d^nx\left[\left(\frac{\partial\phi}{\partial t}\right)^2+|\nabla\phi|^2-\frac{1}{2}\mu^2\phi^2+\lambda\phi^4\right].$$

There are several ways to discretize this model. One can imagine to have a lattice spacing in all directions (we assume an Euclidean metric) and so

$$\partial_\mu\phi(x)=\frac{\phi(x+\hat ia_\mu)-\phi(x)}{a_\mu}$$

and what you will get at the end of computation is a model like (a good reference is this)

$$S=\frac{1}{2}\sum_{xy}\phi(x)M(x,y)\phi(y)$$

for the free part. You should compare this with a generic Ising model

$$S=\sum_{ij}\sigma_iJ_{ij}\sigma_j$$

So, in a small perturbation limit the Ising model and the scalar field theory displays a similar behavior. For the Ising model this corresponds to the mean field approximation.

Similarly, if you take the limit $\lambda\rightarrow\infty$, a strongly coupled scalar field theory, again you recover a Ising model. These two models are said to belong to the same universality class as firstly showed by Kenneth Wilson.

So, from this you can see that the identification with a kinetic or potential energy will depend on the way you are considering your model. But, in a mean field approximation with all its corrections, the correspondence with an almost pure kinetic term is obtained.