Learning Entropy Production from Underdamped Langevin Trajectories

Entropy production (EP) is a central quantity in nonequilibrium physics as it monitors energy dissipation, irreversibility, and free energy differences during thermodynamic transformations and computations. Estimating EP, however, is challenging both theoretically and experimentally due to limited access to the system dynamics. For overdamped Langevin dynamics and Markov jump processes it was recently proposed that, from thermodynamic uncertainty relations (TUR), short-time cumulant currents can be used to estimate EP without knowledge of the dynamics. Yet, estimation of EP in underdamped Langevin systems remains an active challenge, especially for long-time averaging and in nonsteady-state scenarios. To address this, we derive a modified TUR that relates the statistics of two specific novel currents---one cumulant current and one stochastic current---to a system's EP. These two distinct but related currents are used to constrain EP in the modified TUR. One highlight is that there always exists a family of currents such that the uncertainty relations saturate. This uncertainty relation allows estimating EP for both overdamped and underdamped Langevin dynamics. We validate the method numerically, through applications to several underdamped systems, to underscore the flexibility in obtaining EP in nonequilibrium Langevin systems. Additionally, investigation of the cumulant current reveals a connection between supervised machine learning and the estimation problem; this allows for straightforward extensions to estimations of more observables and, potentially, a larger class of dynamics.