Global thermodynamic manifold for conservative control of stochastic systems

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 foundation for this friction-tensor formalism for conservatively driven systems on either discrete- or continuous-state spaces. 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. We leverage a simple mathematical relationship between system dynamics and the geometry of the global manifold to derive expressions for the generalized friction and linear-response excess work. We show that the friction tensor, usually defined as an infinite-time integral, may be expressed as a product of straightforwardly computed matrices. We furthermore show that the linear-response excess power is decomposable into relaxation-time-scaled projections of the control velocity onto eigenmodes of the rate matrix or Fokker-Planck operator. This intuitive result provides a systematic means of interpreting otherwise-mysterious efficient control strategies, unveiling how multiparameter optimal control takes advantage of multiexponential relaxation dynamics to reduce dissipation relative to single-parameter control. We demonstrate the computational and conceptual advances in three illustrative examples.