Thermodynamics and optimal protocols of multidimensional quadratic Brownian systems

Abstract

We characterize finite-time thermodynamic processes of multidimensional quadratic overdamped systems. Analytic expressions are provided for heat, work, and dissipation for any evolution of the system covariance matrix. The Bures-Wasserstein metric between covariance matrices naturally emerges as the local quantifier of dissipation. General principles of how to apply these geometric tools to identify optimal protocols are discussed. Focusing on the relevant slow-driving limit, we show how these results can be used to analyze cases in which the experimental control over the system is partial.

Optimal control with a strong harmonic trap

“…For one-dimensional overdamped dynamics, optimaltransport theory can be directly applied to determine the minimum-dissipation protocol that drives a system between specified initial and final distributions 16 and to place fundamental bounds on the cost of information erasure 17 . For multidimensional overdamped dynamics, the entropy production is lower bounded by the Wasserstein distance between the initial and final distributions 18,19 , and exact minimum-dissipation protocols are known for quadratic trapping potentials in isolation 20,21 .…”

“…Solving the dynamical equation of motion (1) for the time-dependent mean and covariance and substituting their optima (8a) and (8b), the trap center and stiffness must respectively satisfy (for a detailed derivation in the absence of an energy landscape see [21])…”