Optimal active engines obey the thermodynamic Lorentz force law (opens in new tab)

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 coupl...