Multidimensional entropic bound: Estimator of entropy production for Langevin dynamics with an arbitrary time-dependent protocol

Entropy production (EP) is a key quantity in thermodynamics, and yet measuring EP has remained a challenging task. Here we introduce an EP estimator, called multidimensional entropic bound (MEB), utilizing an ensemble of trajectories. The MEB can accurately estimate the EP of overdamped Langevin systems with an arbitrary time-dependent protocol. Moreover, it provides a unified platform to accurately estimate the EP of underdamped Langevin systems under certain conditions. In addition, the MEB is computationally efficient because optimization is unnecessary. We apply our developed estimator to three physical systems driven by time-dependent protocols pertaining to experiments using optical tweezers: A dragged Brownian particle, a pulling process of a harmonic chain, and an unfolding process of an RNA hairpin. Numerical simulations confirm the validity and efficiency of our method.