Geometry of thermodynamic control

A fundamental problem in modern thermodynamics is how a molecular-scale machine performs useful work, while operating away from thermal equilibrium without excessive dissipation. We show that when a thermodynamic system is driven from equilibrium, in the linear response regime, the space of controllable parameters has a Riemannian geometry induced by a generalized friction tensor. This metric structure controls the dissipation of finite-time transformations, and bestows optimal protocols (geodesics on the Riemannian manifold) with many useful properties. We exploit this geometric insight to construct closed-form expressions for minimal-dissipation protocols for a model system of a particle diffusing in a one-dimensional harmonic potential, where the spring constant, inverse temperature, and trap location are adjusted simultaneously. This simple model has a surprisingly rich geometry, which we test via a numerical implementation of the Fokker-Planck equation.