# Thermodynamic uncertainty relation for underdamped Langevin systems driven by a velocity-dependent force

## TL;DR

This paper extends the thermodynamic uncertainty relation (TUR) to underdamped Langevin systems with velocity-dependent forces, providing a more general framework applicable to various physical systems.

## Contribution

The authors derive a generalized TUR for underdamped Langevin dynamics with velocity-dependent forces, expanding the applicability of TUR beyond overdamped systems.

## Key findings

- Accurately describes trade-offs in molecular refrigerators
- Applies to Brownian particles in magnetic fields
- Validates with active particle systems

## Abstract

Recently, it has been shown that there is a trade-off relation between thermodynamic cost and current fluctuations, referred to as the thermodynamic uncertainty relation (TUR). The TUR has been derived for various processes, such as discrete-time Markov jump processes and overdamped Langevin dynamics. For underdamped dynamics, it has recently been reported that some modification is necessary for application of the TUR. In this study, we present a more generalized TUR, applicable to a system driven by a velocity-dependent force in the context of underdamped Langevin dynamics, by extending the theory of Vu and Hasegawa [preprint arXiv:1901.05715]. We show that our TUR accurately describes the trade-off properties of a molecular refrigerator (cold damping), Brownian dynamics in a magnetic field, and an active particle system.