Global thermodynamic manifold for conservative control of stochastic systems

Abstract: Optimal control of stochastic systems plays a central role in nonequilibrium physics, with applications in the study of biological molecular motors and the design of single-molecule experiments. While exact analytic solutions to optimization problems are rare, under slow driving conditions, the problem can be reformulated geometrically solely in terms of equilibrium properties. In this framework, minimum-work protocols are geodesics on a thermodynamic manifold whose metric is a generalized friction tensor. Here, we introduce a new foundation for this friction-tensor formalism for conservatively driven systems. Under complete control of the potential energy, a global thermodynamic manifold (on which points are identified with instantaneous energy landscapes) has as its metric a full-control friction tensor. Arbitrary partial-control friction tensors arise naturally as inherited metrics on submanifolds of this global manifold. Leveraging a simple mathematical relationship between system dynamics and the geometry of the global manifold, we derive new expressions for the friction tensor that offer powerful tools for interpretation and computation of friction tensors and minimum-work protocols. Our results elucidate a connection between relaxation and dissipation in slowly driven systems and suggest optimization heuristics. We demonstrate the utility of these developments in three illustrative examples.