# Two Dimensional Ising Model and Hamiltonian.

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 $U(q)$?

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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":

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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.

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