Optimality of nonconservative driving for finite-time processes with discrete states

Driving rapidly while remaining in control: classical shortcuts from Hamiltonian to stochastic dynamics

Stochastic thermodynamics lays down a broad framework to revisit the venerable concepts of heat, work and entropy production for individual stochastic trajectories of mesoscopic systems. Remarkably, this approach, relying on stochastic equations of motion, introduces time into the description of thermodynamic processes-which opens the way to fine control them. As a result, the field of finite-time thermodynamics of mesoscopic systems has blossomed. In this article, after introducing a few concepts of control for isolated mechanical systems evolving according to deterministic equations of motion, we review the different strategies that have been developed to realize finite-time state-to-state transformations in both over and underdamped regimes, by the proper design of time-dependent control parameters/driving. The systems under study are stochastic, epitomized by a Brownian object immersed in a fluid; they are thus strongly coupled to their environment playing the role of a reservoir. Interestingly, a few of those methods (inverse engineering, counterdiabatic driving, fast-forward) are directly inspired by their counterpart in quantum control. The review also analyzes the control through reservoir engineering. Besides the reachability of a given target state from a known initial state, the question of the optimal path is discussed. Optimality is here defined with respect to a cost function, a subject intimately related to the field of information thermodynamics and the question of speed limit. Another natural extension discussed deals with the connection between arbitrary states or non-equilibrium steady states. This field of control in stochastic thermodynamics enjoys a wealth of applications, ranging from optimal mesoscopic heat engines to population control in biological systems.

Limited-control optimal protocols arbitrarily far from equilibrium

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 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 calculable numerical solutions. Within this framework, we reproduce analytical results for the optimal control of harmonic potentials and numerically devise 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 nonmonotonic in time.

Entropy production given constraints on the energy functions

We consider the problem of driving a finite-state classical system from some initial distribution p to some final distribution p^{'} with vanishing entropy production (EP), under the constraint that the driving protocols can only use some limited set of energy functions E. Assuming no other constraints on the driving protocol, we derive a simple condition that guarantees that such a transformation can be carried out, which is stated in terms of the smallest probabilities in {p,p^{'}} and a graph-theoretic property defined in terms of E. Our results imply that a surprisingly small amount of control over the energy function is sufficient (in particular, any transformation p→p^{'} can be carried out as soon as one can control some one-dimensional parameter of the energy function, e.g., the energy of a single state). We also derive a lower bound on the EP under more general constraints on the transition rates, which is formulated in terms of a convex optimization problem.