Limited-control optimal protocols arbitrarily far from equilibrium

TL;DR

This paper develops exact finite-time optimal control protocols for stochastic thermodynamic systems with limited external control, surpassing previous slow or fast-driving approximations, and demonstrates their effectiveness through analytical and numerical examples.

Contribution

It introduces a novel optimal control framework for limited-control systems far from equilibrium, solving Hamiltonian PDEs for exact protocols and applying them to complex potentials.

Findings

Optimal protocols outperform previous methods.

Mean position travels at near-constant velocity under optimal control.

Optimal protocols can be non-monotonic in time for certain regimes.

Abstract

Recent studies have explored finite-time dissipation-minimizing protocols for stochastic thermodynamic systems driven arbitrarily far from equilibrium, when granted full external control to drive the system. However, in both simulation and experimental contexts, systems often may only be controlled with a limited set of degrees of freedom. Here, going beyond slow- and fast-driving approximations employed in previous studies, we obtain exact finite-time optimal protocols for this unexplored limited-control setting. By working with deterministic Fokker-Planck probability density time evolution, we can frame the work-minimizing protocol problem in the standard form of an optimal control theory problem. We demonstrate that finding the exact optimal protocol is equivalent to solving a system of Hamiltonian partial differential equations, which in many cases admit efficiently calculatable numerical solutions. Within this framework, we reproduce analytical results for the optimal control of harmonic potentials, and numerically devise novel optimal protocols for two anharmonic examples: varying the stiffness of a quartic potential, and linearly biasing a double-well potential. We confirm that these optimal protocols outperform other protocols produced through previous methods, in some cases by a substantial amount. We find that for the linearly biased double-well problem, the mean position under the optimal protocol travels at a near-constant velocity. Surprisingly, for a certain timescale and barrier height regime, the optimal protocol is also non-monotonic in time.