Limited-control optimal protocols arbitrarily far from equilibrium

Abstract

Optimizing the energy efficiency of finite-time processes is of major interest in the study of nonequilibrium systems. 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, often times in both simulation and experimental contexts, systems may only be controlled with a limited set of degrees of freedom. Here, we apply ideas from optimal control theory to obtain optimal protocols arbitrarily far from equilibrium for this unexplored limited-control setting.By working with deterministic Fokker-Planck probability distribution time evolution and using the first law of thermodynamics to recast the work-expenditure, 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 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 significant amount. We find that for the linearly biased double-well problem, the mean position under the optimal protocol travels at a near-constant velocity, and that surprisingly, for a certain timescale and barrier height regime, the optimal protocol is non-monotonic in time.

Thermodynamics and optimal protocols of multidimensional quadratic Brownian systems

“…This means that fixing the boundary values of Σ is non-trivially related to fixing boundary controls. The results of optimisation thus strongly depend on the imposed constraints [65,66,67]. At the same time, for the purpose of typical applications to isothermal processes (cf section 5), in which the goal is to minimize work dissipation, we can consider the slow-driving limit of the dynamics [68].…”