A Markov decision process formulation of the low-temperature Ising model with Kawasaki dynamics on a square lattice reveals that optimal growth strategies depend entirely on the reward structure. When the objective is speed --- reaching the all-occupied target state as fast as possible --- optimal policies favor adding particles at cluster boundaries, maximizing the rate of interface advance. When the objective accounts for energy cost, optimal policies shift to corner attachment, exploiting the higher binding energy at corners to minimize thermodynamic expenditure per particle added. The two strategies are not minor variations; they represent qualitatively different geometric growth modes emerging from the same underlying dynamics.
Metastable nucleation has been studied for decades through the lens of free energy barriers and critical droplet theory, but framing the controller as an external agent with an objective function reframes the entire problem. The cluster is no longer a passive statistical object governed by detailed balance; it is a controllable system whose growth pathway can be steered. The boundary-versus-corner distinction maps directly onto a fundamental tension in growth problems: fast growth exploits high-coordination sites, but energetically efficient growth exploits high-binding sites, and in a lattice geometry these are different locations. The reward structure does not merely select between strategies --- it selects between geometries.
Optimization reveals the hidden degrees of freedom in processes that appear deterministic from a thermodynamic perspective. A nucleation event looks like a single pathway when viewed through free energy; viewed through optimal control, it is a decision tree where every attachment site represents a choice with distinct time and energy consequences. The general principle: any physical process that appears to follow a unique pathway under one objective function may bifurcate into qualitatively different pathways under another, and the structure of the bifurcation encodes the geometry of the tradeoff space.
(arXiv:2603.18903)