Generalized Finite-time Optimal Control Framework in Stochastic Thermodynamics

TL;DR

The paper addresses finite-time, thermodynamically efficient control of discrete-state stochastic systems far from equilibrium by formulating GFTOC, a minimum-action principle–based variational framework. It unifies slow-driving thermodynamic geometry and finite-time optimal control through an exact mapping, and introduces a boundary term concept—the thermodynamic shock—that accounts for endpoint jumps (kinks) in finite-time protocols. Exact slow- and finite-time geodesics are derived, with a clear decomposition of dissipation into bulk and boundary contributions, and a universal relation to Wasserstein distance in the slow limit. The framework extends to multi-parameter controls and provides concrete applications, including linear, free-energy, and Gaussian-distributed systems, showing broad implications for designing thermodynamically efficient protocols in nanoscale and mesoscale devices.

Abstract

Optimal processes in stochastic thermodynamics are a frontier for understanding the control and design of non-equilibrium systems, with broad practical applications in biology, chemistry, and nanoscale/mesoscale systems. Optimal mass transport theory and thermodynamic geometry have emerged as optimal control methodology, but they are based on slow-driving and close to equilibrium assumptions. An optimal control framework in stochastic thermodynamics for finite time driving is still elusive. Therefore, we solve in this paper an optimal control problem for changing the control parameters of a discrete-state far-from-equilibrium process from an initial to a final value in finite-time. Optimal driving protocols are derived that minimize the total finite-time dissipation cost for the driving process. Our framework reveals that discontinuous endpoint jumps are a generic, model-independent physical mechanism that minimizes the optimal driving entropy production, whose importance is further amplified for far-from-equilibrium systems. The thermodynamic and dynamic physical interpretation and understanding of discontinuous endpoint jumps is formulated. An exact mapping between the finite-time to slow driving optimal control formulation is elucidated, developing the state-of-the-art of optimal mass transport theory and thermodynamic geometry, which has been the current paradigm for studying optimal processes in stochastic thermodynamics that relies on slow driving assumptions. Our framework opens up a plethora of applications to the thermodynamically efficient control of a far-from-equilibrium system in finite-time, which opens up a way to their efficient design principles.

Figures (2)

• Figure 1: (a) The curvature of the Lagrangian $\partial_\alpha^2 \mathcal{L}^*$ (mass) increases exponentially for the higher value of driving affinity $A_\alpha$. (b) $\mathcal{G}(F_\alpha): F_\alpha \to t$. Comparison between $\mathcal{G}^{cEQ}(F_\alpha)$, $\mathcal{G}^{fEQ}(F_\alpha)$ and $\mathcal{G}^{lin}(F_\alpha)$ given by \ref{['eq:geodesic_expressions']}. For the fixed quasi-static driving speed $v_{qs}$, $F_\alpha^f - F_\alpha^i = 1$ is considered for the cEQ $F_\alpha^i=1$ and fEQ $F_\alpha^i = 3.5$ and the corresponding $\Delta \tau_c$ and $\Delta \tau_f$ are plotted; it is a combined pictorial visualization of \ref{['eq:geodesic_expressions', 'eq:slow_driving_speed_limit']}. Due to \ref{['eq:slow_driving_speed_limit']}, the speed limit for the quasi-static slow-driving increases for fEQ systems, an effect attributed to the higher mass required for the driving of fEQ systems.
• Figure 2: (a) The finite-time optimal protocol $\mathcal{G}_{\tau}(F_\alpha) \to t$ is plotted for the different values of $\tau = \{4, 1, 0.1\}$ with the same initial and final value condition (shown by the dotted blue lines) with $F_\alpha^i = 2$ and $F_\alpha^f = 5$. Its slow-driving counterpart is denoted by 'qs'. The vanishing excess affinity is considered because the optimal control of the housekeeping EPR is being solved. Where, \ref{['eq:optimal_protocol', 'eq:optimal_geodesic_finite_time_explicit', 'eq:geodesic_expressions']} are used for the plot and the scaled (with $\tau$) driving time ($t/\tau$) is used. Kinks are of equal amplitude in the geodesic space. (b) The corresponding finite-time non-conservative affinity is obtained using the fEQ geodesic $\mathcal{G}^{fEQ}(F_\alpha)$ in \ref{['eq:geodesic_expressions']}, which gives the mapping $F_\alpha \to t$ in scaled (with $\tau$) driving time ($t/\tau$). Kinks are of unequal amplitude in the affinity space, such that a higher mass point exhibits a smaller amplitude of kinks. (c) The corresponding mapping in $\mathcal{G}_{\tau}(F_\alpha) \to F_\alpha$ space, the initial and final points are denoted by $\star$ and $\blacklozenge$, respectively. Due to the finite driving time constraint, the finite-time optimal protocol \ref{['eq:optimal_geodesic_finite_time_explicit']} traverses a part of the slow-driving geodesic \ref{['eq:optimal_protocol']}, such that the finite-time speed limit \ref{['eq:slow_driving_speed_limit_finite_time']} is restored, analogous to the slow-driving speed limit \ref{['eq:slow_driving_speed_limit']}. The choice of discontinuous jumps in the finite-time driving protocol is also constrained by equally distributing the thermodynamic cost ($\Sigma_{bnd}^*$ in \ref{['eq:optimal_finite_time_bulk_boundary_EP']}) between the initial and final endpoints, which generates a 'Thermodynamic shock' at the initial and final time.