Mapping phase diagrams of colloidal systems is laborious. You simulate or experimentally probe each state point, wait for equilibrium, characterize the structure, and repeat across parameter space. A system with ten possible phases means you need enough sampling to find all ten — and to find the boundaries between them.

The dispersion relation tells you where to look.

The key insight is that ω(k) from dynamical density functional theory — the relation governing whether density fluctuations at wavenumber k grow or decay — directly predicts where complex crystal phases form. Regions of the phase diagram where specific wavenumbers become unstable correspond to regions where the crystalline phase characterized by those wavenumbers actually appears. The instabilities are the map.

For a system of particles with core-shoulder pair interactions, this approach identifies at least ten distinct phases, including quasicrystals. By tuning the interaction potential to control which wavenumbers become unstable, the researchers can design systems with desired crystal structures. The dispersion relation doesn't just predict where known phases live — it reveals what phases are accessible.

This inverts the usual relationship between stability analysis and phase behavior. Stability analysis is typically a local tool: given a state, determine whether small perturbations grow or decay. Here, the local analysis generates the global map. The pattern of instabilities across parameter space is isomorphic to the phase diagram itself — not an approximation of it, not a hint toward it, but a direct representation that accelerates the exploration by orders of magnitude.

The phase diagram was always encoded in the dynamics. We were solving a cartography problem with the tools of equilibrium, when the dispersion relation had already drawn the boundaries.