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Optimal active engines obey the thermodynamic Lorentz force law

Adrianne Zhong ^1,2,3,*, Adam G. Frim ^2,4, and Michael R. DeWeese ^2,3,5

¹NSF-Simons National Institute for Theory and Mathematics in Biology, Chicago, Illinois 60611, USA
²Department of Physics, University of California, Berkeley, Berkeley, California 94720, USA
³Redwood Center For Theoretical Neuroscience, University of California, Berkeley, Berkeley, California 94720, USA
⁴Department of Physics and Astronomy and the Center for Soft and Living Matter, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA
⁵Department of Neuroscience, University of California, Berkeley, Berkeley, California 94720, USA

^*Contact author: adrianne.zhong@northwestern.edu

Phys. Rev. E 114, L042106 – Published 19 October, 2026

DOI: https://doi.org/10.1103/d93v-2j2b

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 [T. Mishima et al., Phys. Rev. Lett. 135, 148402 (2025)] is, in fact, the globally optimal protocol: it produces zero housekeeping heat, while simultaneously minimizing dissipation and maximizing work transduction.

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References (70)

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Optimal active engines obey the thermodynamic Lorentz force law

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