Describe Ising model dynamics in stochastic differential equation or stochastic process

Question

The Ising model is described by the Hamiltonian $$ H(\sigma) = - \sum_{<i~j>} J_{ij} \sigma_i \sigma_j -\mu \sum_{j} h_j\sigma_j, $$ and is treated extensively by equilibrium statistical mechanics. Consider a Ising model at equilibrium is affected by a sudden change in external parameters such as the temperature and the external field and the system goes to a new equilibrium state after some time. Monte Carlo simulation can simulate this non-equilibrium process, but can this relaxation from a non-equilibrium state to a equilibrium state be described by a system of coupled differential equation or a stochastic process?

Answer 1

I'm not a 100% this will address the question, so this might be more of a long comment than an answer. My main point is, perhaps, that given a Hamiltonian, you'll still want to specify the dynamics to do simulations. Even the Ising model can be simulated in several different ways, all satisfying detailed balance, where the relaxation towards equilibrium is different (as is evident from the differences in Metropolis/Swendsen-Wang/Wolff etc.). What is correct, then, depends on what you're trying to achieve.

Lattice models

Normally in the context of the Ising model, one thinks in terms of magnetic spins, and there is no absolute restriction as to the total magnetization: If you are below the critical temperature, the symmetry (if there is no external field, that is) will be spontaneously broken and you'll end up with a system filled with either up or down spins. Fine. This type of simulation model where you pick a spin and stochastically flip it according to some energy condition is sometimes called Glauber dynamics.

Suppose you were talking about the lattice gas, instead. This model can be mapped to Ising: Essentially you'll have the up spins represent particles and the down spins empty sites. Now if the particle-particle interaction is strong enough (or the temperature low enough), what you get is fluid-gas phase separation: All the particles clump together. Indeed, you can even compute the surface tension of such a fluid. Now if you simulate this according to the Glauber rules, your entire simulation will be filled with either particles or empty sites, as established above. If we fill the entire box with particles, there is no interface whose tension we might compute; we will only have bulk. Right, so obviously particles are rushing in from the nefarious environment, or bath, if you will. But we don't want this. Enter Kawasaki dynamics, in which you never flip a spin, but rather you pick two sites at random and exchange them according to some energy condition. This scheme is guaranteed to keep the global order parameter (total magnetization, particle number, or whatever it may be) constant.

But this all you probably knew. Let's get to the continuum stuff where stochastic differential equations enter the picture.

Continuum models

Take the Ising Hamiltonian and massage it a bit in the spirit of Ginzburg-Landau:

$$\mathcal{F}[\varphi(\mathbf{r}, t)] = f(\varphi(\mathbf{r}, t)) + \frac{k}{2}(\nabla \varphi(\mathbf{r}, t))^2$$

where the first term $f$ is the local phenomenological free energy, and $\varphi$ the local order parameter. What local means, like usually in continuum mechanics, is not one spin but a small volume of spins. Usually such a volume might be well described by the mean field approximation, perhaps something of the form $f(\varphi) = \varphi^4 - \varphi^2$ if one Taylor expands the $\log$s.

Now how do we simulate such a system? Well obviously the "ground state" is determined by the Euler-Lagrange equations $\frac{\delta \mathcal{F}}{\delta \varphi} = 0$. The simplest way to actually do this numerically then, would be to slowly step towards this state, i.e.

$$\frac{\partial\varphi(\mathbf{r}, t)}{\partial t} = -\Gamma\frac{\delta\mathcal{F}}{\delta \varphi(\mathbf{r}, t)} + \xi(\mathbf{r}, t)$$

where I've thrown $\xi$, random noise, in for good measure. This should satisfy the fluctuation-dissipation relation if we're being physical. So that's it, a simple scheme to relax towards equilibrium. That's model A. The most typical example is the Allen-Cahn model.

Clearly model A does not conserve the total order parameter: the change in order parameter gets larger the smaller the relative chemical potential ($\frac{\delta\mathcal{F}}{\delta \varphi}$) is. The order parameter might, and probably will, eventually become uniform, just like in the regular Ising model all the spins point to one direction with Glauber dynamics. This is no good if we want to simulate particles whose total number is conserved. Well, we could just fix the boundary conditions so as to always force an interface to exist in the system if interfaces and their tensions is what we're interested in.

What model B does instead, though, is to force the conservation of the global order parameter. It does this simply by stating that the order parameter does not locally just respond to the chemical potential, but rather that it can only diffuse (to areas with a lower chemical potential). Sound complicated? It's simpler in equations:

$$\frac{\partial\varphi(\mathbf{r}, t)}{\partial t} = \lambda\nabla^2\frac{\delta\mathcal{F}}{\delta \varphi(\mathbf{r}, t)} + \xi(\mathbf{r}, t)$$

Note that the Cahn-Hilliard model follows this form. Kawasaki dynamics is very similar to model B, although in Kawasaki one can exchange spins between far away sites (whereas in model B the order parameter diffuses to "nearby" sites; arguably the kernel is infinite, though). There we go, we have now "derived" the most common of the phenomenological equations referred to as time-dependent Ginzburg-Landau. While there are several others (see for example Chaikin & Lubensky Principles of Condensed Matter Physics p. 467) the one worth mentioning is model H (I always thought H stood for hydrodynamics, but in going through some references just now, it occurred to me that it might refer to the authors' names, Hohenberg & Halperin). Instead of just $\varphi$, one would also model the flow field $\mathbf{v}$: $$(\partial_t + \mathbf{v}\cdot\nabla)\varphi(\mathbf{r}, t) = \lambda \nabla^2 \frac{\delta \mathcal{F}}{\delta \varphi(\mathbf{r}, t)} + \xi(\mathbf{r}, t)$$ $$v_i(\mathbf{r}, t) = \int \mathrm{d}\mathbf{r}' T_{ij}(\mathbf{r}-\mathbf{r}')\left(\frac{\delta \mathcal{F}}{\delta \varphi(\mathbf{r}', t)}\nabla'_j\varphi(\mathbf{r}', t) + \zeta_j(\mathbf{r}', t)\right)$$ where $T_{ij}$ is the Oseen tensor (or whatever your flow dynamics dictate).

I am not quite sure how one would go about simulating these types of dynamics with a lattice model similar to Ising. I suppose you might have to use some lattice gas automata (or lattice Boltzmann, but I don't count that as a "proper" lattice method in that it uses real numbers and thus I think does not map to a finite state machine).

Do note that there are several subtleties if you decide to solve these models (numerically, though some equilibrium properties can be derived analytically), in particular with regards to noise (one must take care in noting how the fluctuation-dissipation relation is defined and also how additive noise works/Itô-Stratonovich dilemma). My favorite method of doing the actual simulation, if you care, is the semi-implicit Fourier space method, whereby your linear terms are evaluated on the "next time step" and the nonlinear "this time step".

Answer 2

I would like to modestly add that if one calculates the mean radius of the magnetic domains in a Ising lattice relaxing to equilibrium below the critical temperature, then this radius is empirically found to be proportional to the cubic root of time. In fact, this cubic root dependence is listed in the table of critical scaling laws of the Ising class in textbooks (and I think in wikipedia).

Fortunately, the cubic root dependence can be analytically derived from model A, which thus provides a way to quantitatively compare these phase field approaches or if you wish "computational show" their consistency. However, based on the references and discussions I have seen, I do not think it is possible to rigorously derive these formulations starting from lattice Hamiltonian and the microscopic dynamics.