Universal form of thermodynamic uncertainty relation for Langevin dynamics

TL;DR

This paper derives a universal thermodynamic uncertainty relation applicable to all Langevin dynamics, including underdamped systems with odd parity variables, unifying previous results and confirming its validity through multiple prototypical models.

Contribution

The authors present a new, operationally accessible TUR for underdamped Langevin dynamics that reduces to the original TUR in the zero-mass limit, establishing a universal form.

Findings

Validated for free diffusion, charged Brownian particle, and molecular refrigerator.

Shows the TUR reduces to the original form in the overdamped limit.

Provides a unified framework for TUR in general Langevin systems.

Abstract

Thermodynamic uncertainty relation (TUR) provides a stricter bound for entropy production (EP) than that of the thermodynamic second law. This stricter bound can be utilized to infer the EP and derive other trade-off relations. Though the validity of the TUR has been verified in various stochastic systems, its application to general Langevin dynamics has not been successful in a unified way, especially for underdamped Langevin dynamics, where odd parity variables in time-reversal operation such as velocity get involved. Previous TURs for underdamped Langevin dynamics is neither experimentally accessible nor reduced to the original form of the overdamped Langevin dynamics in the zero-mass limit. Here, we find an operationally accessible TUR for underdamped Langevin dynamics with an arbitrary time-dependent protocol. We show that the original TUR is a consequence of our underdamped TUR in the zero-mass limit. This indicates that the TUR formulation presented here can be regarded as the universal form of the TUR for general Langevin dynamics. The validity of our result is examined and confirmed for three prototypical underdamped Langevin systems and their zero-mass limits; free diffusion dynamics, charged Brownian particle in a magnetic field, and molecular refrigerator.