Exploring the Ising model and the emergence of collective order from local interactions.
A hot magnet is a mess of jiggling atoms. Cool it past a critical temperature and the atoms align into a global magnetic field — a phase transition.
The Ising model is the standard tool for studying these transitions: a lattice of up/down spins where neighbors prefer to align and thermal energy randomizes them. The balance is set by temperature $T$.
At $T_c$ the system is poised between order and disorder. Clusters of aligned spins form at every size, from a few atoms to the whole lattice. These critical fluctuations are scale-invariant.
Figure 1. A 2D Ising model simulation. At the critical temperature \(T_c \approx 2.269\), notice how the patterns of blue and orange become fractal-like, with clusters of all sizes.
The lattice is the microscopic view. A temperature sweep turns those flickering configurations into macroscopic curves: the expected magnetization $|M|$ stays large below $T_c$, while susceptibility $\chi$ peaks where domain-scale fluctuations are easiest to excite.
Figure 2. A sweep through temperature. The blue curve tracks expected order, the green curve tracks fluctuation response, and the small lattice samples show why the susceptibility peak corresponds to the large clusters in Figure 1.
The same dynamics drive phase transitions in any system where local imitation competes with noise: bird flocks, opinion dynamics, neural avalanches.
Different systems often share the same critical exponents. Near the transition, magnetization vanishes as $M \propto (T_c - T)^\beta$. Only the dimensionality and symmetry of the interactions determine $\beta$ and the other exponents — agent details drop out.