A global thermodynamic manifold for optimal conservative control

The problem of determining minimum-work protocols (MWPs) for driving stochastic systems arises in many contexts, including the design of single-molecule experiments, the study of biological molecular motors, and in the development of nanotechnology. A widely used approximation recasts MWPs as geodesics on a thermodynamic manifold whose metric tensor is a generalized friction that depends only on a system's equilibrium properties. We advance this framework by introducing a full-control tensor that describes a global thermodynamic manifold from which arbitrary conservative-control friction tensors may be obtained as inherited metrics on hypersurfaces of constraint. This geometric structure offers a practical means of systematically comparing different sets of control parameters. Computation is also facilitated in this framework: while parametric-control friction tensors have typically been estimated from simulations, we show that the full-control tensor is obtainable from elementary matrix operations. Additionally, a spectral decomposition of the friction-tensor suggests design principles for MWPs and provides insight into the domain of applicability of the formalism.