The cost of speed: Time-optimal thermal control of trapped Brownian particles

Abstract

A thermal analogue of the classical brachistochrone problem, which minimizes the connection time between two equilibrium states of harmonically confined Brownian particles, has recently been solved theoretically. Here we report its experimental realization using two optically trapped microparticles subjected to a bang-bang effective temperature protocol. Despite their distinct relaxation times, both degrees of freedom are steered to their respective equilibrium states simultaneously in a finite minimal time. We provide a complete time-resolved characterization of the nonequilibrium dynamics through the evolution of the position variances and the entropy production within the framework of stochastic thermodynamics, enabling a quantitative comparison with direct relaxation and a suboptimal protocol. In addition, we employ information-geometric tools -- recently referred to as thermal kinematics -- to track the system's path in state space with a single dynamical quantity. Our results show that faster equilibration requires a larger entropy production and an increased thermodynamic length, revealing a direct trade-off between temporal optimality and thermodynamic cost in multidimensional stochastic systems driven by a single intensive control parameter.

Figures (4)

• Figure 1: Thermal brachistochrone and experimental platform.(a) We connect two equilibrium states by changing only the bath temperature while keeping the trap stiffnesses fixed. A direct quench from $T_0$ to $T_{\rm f}$ (top) relaxes exponentially. The time-optimal thermal brachistochrone (bottom) is a bang-bang protocol that switches between the accessible bounds $T_{\min}$ and $T_{\max}$ and reaches the target state in the minimum possible time for the given values of $\{T_{\min},T_{\max}\}$. (b) Two charged dielectric microspheres dispersed in water are independently confined in harmonic optical traps generated by a split, tightly focused laser beam. The trap centers are separated by a distance $d$ much larger than the particle diameter, so hydrodynamic coupling is negligible. An external, spatially uniform noisy electric field produced by submerged microelectrodes drives the particles and sets an effective bath temperature $T$ above the ambient value, yielding Brownian motion with a Gaussian position distribution in each trap at temperature $T$.
• Figure 2: Two-particle variance dynamics under heating protocols connecting the same equilibrium states.(a) Bath temperature $T(t)$ for the three heating protocols applied to the same two-particle system: direct relaxation (purple), suboptimal one-bang protocol (green), and optimal two-bang (brachistochrone) protocol (red). (b) and (c) Corresponding rescaled variances $\widetilde{s}_i(t)$ for the slow (with a characteristic time $\tau_{\rm slow} = 1.35 \ {\rm ms}$) and fast ($\tau_{\rm fast} = 0.72 \ {\rm ms}$) particles. Vertical dashed lines mark the instants at which the temperature jumps in each protocol, with the same color code. Symbols are experimental data and solid lines are the theoretical predictions without fitting parameters. Error bars denote the standard error of the mean over 11400 repeated runs of each protocol. Gray curves correspond to the direct relaxation $T_0\to T_{\max}$ shown as a reference.
• Figure 3: Thermal-kinematics characterization of the two-particle heating protocols. Degree of completion $\varphi(t)=\mathcal{L}(t)/\mathcal{L}_{\rm tot}$ for the full two-particle system under direct relaxation (purple), the suboptimal protocol (green), and the optimal (brachistochrone) protocol (red), for the same data set as Fig. \ref{['fig:fig2']}. Again, symbols are experimental data and solid lines are the corresponding theoretical predictions with no fitting parameters. Vertical dashed lines mark the instants at which the temperature jumps during the implementation of each protocol, with the same color code. Error bars are obtained by quadratic propagation of the experimental uncertainties of the variances. Inset: Accumulated thermodynamic length $\mathcal{L}(t)$ as a function of time for the same protocols and color code, showing that the optimal protocol sweeps a longer distance in distribution space in a shorter time.
• Figure 4: Entropic cost for two-particle heating protocols. Total entropy production $\Sigma_{\rm tot}$ as a function of the system entropy change for the direct protocol (purple), the suboptimal protocol (green), and the optimal (brachistochrone) protocol (red). Each symbol corresponds to a distinct experimental series (different control bounds, endpoint temperatures, and stiffness ratios), while data for the same $\Sigma_{\rm sys}$ correspond to the different protocols for the same initial and final states. The hierarchy given by Eq. \ref{['eq:inequality']} is always fulfilled.