Analytical solution for optimal protocols of weak drivings

TL;DR

The paper derives a universal analytical solution for optimal isothermal, finite-time protocols that minimize irreversible work and its variance in weak drivings. By exploiting time-reversal symmetry, the author reduces the problem to a closed-form expression where the optimal protocol comprises a linear time segment plus Dirac-delta and derivative-delta impulses at the boundaries. The approach is validated across multiple relaxation functions (overdamped/underdamped Brownian motion, Sinc/Gaussian/Bessel relaxations), reproducing known results and revealing a common structural pattern: a continuous part $g^*_C(t)=(t+\tau_R)/(\tau+2\tau_R)$ and a singular part built from impulse terms. The work also analyzes asymptotic limits, derives expressions for the minimal irreversible work and its variance, and discusses numerical challenges in solving in the distribution space, highlighting the practical significance for precise control of microscopic thermodynamic processes.

Abstract

One of the main objectives of science is the recognition of a general pattern in a particular phenomenon in some particular regime. In this work, this is achieved with the analytical expression for the optimal protocol that minimizes the thermodynamic work and its variance for finite-time, isothermal, and weak processes. The method that solves the Euler-Lagrange integral equation is quite general and depends only on the time-reversal symmetry of the optimal protocol, which is proven generically for the regime considered. The solution is composed of a straight line with jumps at the boundaries and impulse-like terms. Already known results are deduced, and many new examples are solved corroborating this pattern. Slowly-varying and sudden cases are deduced in their appropriate asymptotic limits. Comparison with numerical procedures is limited by the nonavailability of the present methods of the literature to produce solutions in the space of distributions.