Phase separation in physics

I am only familiar with the rigorous results about the equilibrium description of phase separation in classical lattice spin systems. To be specific (and because it is by and large the only model analyzed in detail), I'll focus on the ferromagnetic Ising model on $mathbb{Z}^d$.

I'll mainly discuss the two-dimensional case, but similar results are also available in higher dimensions (they are not as strong; in particular the topology used is weaker).

So, let us consider the 2d nearest-neighbor ferromagnetic Ising model in the box $Lambda_N={-N,dots, N}^2$ with $+$ boundary condition, at inverse temperature $beta>beta_c$. Denote by $mu^+_{Lambda_n;beta}$ the corresponding Gibbs measure.

In order to describe typical configurations, it is useful to consider the corresponding Peierls contours, that is, the closed lines separating regions of $+$ and $-$:

It is well-known that, for all $beta>beta_c$, typical configurations only contain small contours, in the sense that there exists $K=K(beta)$ such that $$ lim_{ntoinfty} mu_{Lambda_n;beta}^+(text{all contours have diameter < }Klog n) = 1. $$ So, a typical configuration consists in a "sea" of $+$ spins with only small "islands" of $-$.

Let us now turn to a more relevant setting to discuss phase separation: the model with fixed magnetization (that is, the canonical ensemble in the lattice gas interpretation of the Ising model). Namely, let us fix $min[0,1)$ and consider the probability measure $$ mu^+_{Lambda_n;beta,m} (cdot) = mu^+_{Lambda_n;beta,m} Bigl(cdot Bigmvert sum_{iinLambda_n} sigma_i = m|Lambda_n|Bigr). $$ That is, we restrict the measure to configurations with fixed magnetization density $m$. (Of course, I am assuming that $m$ has been chosen as a possible value of the magnetization density in $Lambda_n$.)

One can then show the following. Denote by $m^*_beta$ the spontaneous magnetization density. Then:

• If $mgeq m^*_beta$, then typical configuration still consist in a "sea" of $+$ spins with only small "islands" of $-$. The size of the contours decreases as $m$ increases, in a homogeneous way.
• If $min[0,m^*_beta)$, phase separation occurs: a macroscopic droplet of $-$ spins appears inside the $+$ phase. This droplet has a diameter of order $n$, while all other contours of the configuration are at most of size $Klog n$. Moreover, the shape of this droplet becomes deterministic as $ntoinfty$. Namely, if you let $ntoinfty$ and take a continuum limit (that is, let the mesh size of the lattice tend to $0$ as $1/(2n)$ so that the rescaled box $frac1{2n}Lambda_n$ converges to the unit square), then the shape of the rescaled droplet converges to the shape that minimizes surface tension. (See this answer for the relevant microscopic definition of the surface tension.) Here is a typical configuration ($-$ spin are black, $+$ spins are white):

The volume of the droplet is exactly what it should be to realize the desired magnetization density $m$, given that the magnetization density outside the droplet is $m^*_beta$, while the magnetization density inside the droplet is $-m^*_beta$.

One can actually say much more:

• One can see what happens if one lets $muparrow m^*_beta$ as $ntoinfty$. Then phase separation only occurs if the rate of appraoch is not too fast (extremely precise results about that are known). See this paper for example.

• One can replace the $+$ boundary condition by a boundary magnetic field, which allows one to analyze the wetting transition in this setting: depending on the value of this boundary field, the macroscopic droplet attaches to the boundary. See this review paper.

But I'll stop here, as this answer is long enough. See the above-mentioned review paper for more information, or Section 4.12.1 of this book for a more informal discussion.