Optimal active engines obey the thermodynamic Lorentz force law

TL;DR

This paper establishes a thermodynamic Lorentz force law framework for finite-time work extraction in active nonequilibrium systems, revealing optimal protocols that minimize dissipation and housekeeping heat.

Contribution

It introduces a Lorentz force Lagrangian approach to identify optimal protocols in nonequilibrium thermodynamics, linking electromagnetism concepts to thermodynamic state control.

Findings

Optimal protocols counterdiabatically steer thermodynamic states.

Housekeeping heat can be minimized to near zero.

Constant-velocity angle clamp protocol is globally optimal.

Abstract

What are the fundamental limitations for finite-time engines that extract work from active nonequilibrium systems, and what are the optimal protocols that approach them? We show that the finite-time work extraction for nonconservative overdamped Langevin systems may be rewritten as a Lorentz force Lagrangian action, with the kinetic term corresponding to a thermodynamic metric term that is an $L_2$-optimal transport cost for the time-dependent probability density, and the magnetic field coupling term corresponding to an effective quasistatic work extraction, proving that optimal protocols counterdiabatically steer the thermodynamic state trajectory to satisfy a Lorentz force law defined on thermodynamic state space. We utilize and reinterpret classic concepts from electromagnetism in the setting of cyclical nonequilibrium processes. We show that the housekeeping heat can be controlled to be arbitrarily close to zero by minimizing nonequilibrium fluctuations. It immediately follows from our results that the constant-velocity angle clamp protocol applied to the $F_1$ molecular motor in a recent experiment [Mishima et at, 2025] is in fact the globally optimal protocol: it produces zero housekeeping heat and simultaneously minimizes dissipation and maximizes work transduction.