Thermodynamic Length and Optimal Protocols in Finite-Time Processes

Abstract

Finite-time thermodynamic protocols that drive a system between two equilibrium states inevitably dissipate more
work than the reversible minimum, and the geometric framework of thermodynamic length predicts that this excess
dissipation is minimized by protocols that traverse control-parameter space at constant thermodynamic speed along
the shortest path under a friction-tensor-defined metric. We test this prediction computationally in an overdamped
colloidal-particle-in-a-trap system and a two-dimensional Ising-like spin model, comparing constant-speed geodesic
protocols against naive linear-interpolation protocols across 36 total protocol durations (0.1 to 20 relaxation times)
and, for the two-parameter trap system, across 18 distinct start-end control-point pairs. For the single-parameter
trap-stiffness protocol, geodesic (constant-thermodynamic-speed) protocols reduced mean dissipated work by 34%
relative to linear protocols at short duration (0.1 relaxation times) and by 8% at long duration (5 relaxation times),
consistent with the theoretically predicted 1/duration scaling of the geodesic advantage. For the two-parameter trap
system, where both stiffness and center position were varied simultaneously, the geodesic advantage depended
strongly on the curvature of the underlying friction-tensor metric between the chosen endpoints, ranging from a 46%
reduction in dissipated work for the most curved of the 18 tested endpoint pairs to only 6% for the least curved pair,
with the reduction correlating with the numerically estimated path curvature (Spearman rho = 0.79, p < 0.001). In
the Ising-like spin model, where the control parameter was an external field driving magnetization reversal,
geodesic protocols reduced mean dissipated work by 22% at intermediate duration (1 relaxation time) across 400
independent realizations per protocol type, with the reduction robust to a 3-fold change in system size (L = 20 to L =
60). Measured dissipated work agreed with the thermodynamic-length prediction, W_diss = (L^2)/(2*duration), to
within 11% averaged across all tested durations for the single-parameter trap case. These results provide
quantitative, cross-system validation of the thermodynamic length framework as a practical tool for designing
reduced-dissipation finite-time protocols, and identify path curvature as the dominant factor governing how much
benefit geodesic optimization provides in multi-parameter control spaces.