Thermodynamic control of condensed matter: Driving equilibrium landscapes

We develop a general theoretical framework for the optimal control of equilibrium landscapes governed by arbitrary free energy functionals, mobilities, and multiple control parameters. In the near–quasi-static and weak-noise regime, we show that the minimal energetic cost of driving a passive system between steady states is determined by a thermodynamic metric tensor. This tensor depends solely on the system’s mobility and on how its equilibrium configurations vary with control parameters, thus providing a phenomenological measure of energy efficiency that is agnostic to microscopic interactions. We apply this framework to two paradigmatic problems: (i) energy-optimal reversal of magnetization in interacting spin systems, and (ii) controlled tuning of domain sizes in phase-separating liquids. Our results reveal how symmetries and physical constraints shape the navigable control space, thereby guiding the design of experimentally feasible, energy-efficient protocols. The framework is broadly accessible and can be deployed in diverse soft and condensed matter systems.