q-state Potts Model on a 2D Square Lattice

What you are seeing: a generalization of the Ising model where each site holds one of $q$ colors (instead of just two spin values). Like-colored neighbors bind with energy $-J\,\delta(s_i, s_j)$ ; the system minimizes this energy by forming large monochromatic patches at low temperature and disordering into salt-and-pepper noise at high temperature.

The transition temperature is $T_c(q) = 1 / \ln(1 + \sqrt q)$ . For $q = 2$ this reduces to the Ising model and the transition is second-order: the order parameter $M$ drops smoothly to zero at $T_c$ . For $q \geq 5$ the transition becomes first-order and you see latent-heat jumps in the energy across $T_c$ . The $q = 3$ and $q = 4$ cases sit in between with a marginal second-order behavior.

Below: the left panel is the current spin configuration; the right panel is the order parameter $M(t)$ trace (red horizontal: zero; yellow: current value).

Figure 1. Single-spin Metropolis on the q-state Potts model. Order parameter

M = (q\,n_{\max} - N) / ((q - 1) N)

q 3

T / T_c 1.00

speed 2.0

WHAT TO TRY

Raise q, the number of colors: with more states to disorder into the transition sharpens, and past q = 4 in two dimensions it turns first-order, jumping discontinuously instead of growing smoothly.
Cross T / T_c: above it the lattice is a fizz of random colors, below it one color wins and large monochrome domains lock in, spontaneous symmetry breaking.
Sit right at T_c and watch domains of every size flicker: critical fluctuations are scale-free, the self-similarity that links the Potts model to percolation and to real magnets.