Free diffusion bounds the precision of currents in underdamped dynamics

TL;DR

This paper investigates the limitations of thermodynamic uncertainty relations in underdamped systems, proposing a new bound for driven diffusion that aligns with known results in the overdamped limit and extends to higher dimensions.

Contribution

It introduces a conjectured bound for underdamped driven diffusion that generalizes the overdamped TUR and applies to higher-dimensional systems.

Findings

Underdamped free diffusion violates overdamped TUR at finite times.

A new bound for 1D driven diffusion is proposed, converging to the overdamped TUR.

The bound applies to certain velocity observables and higher-dimensional systems.

Abstract

The putative generalization of the thermodynamic uncertainty relation (TUR) to underdamped dynamics is still an open problem. So far, bounds that have been derived for such a dynamics are not particularly transparent and they do not converge to the known TUR in the overdamped limit. Furthermore, it was found that there are restrictions for a TUR to hold such as the absence of a magnetic field. In this article we first analyze the properties of driven free diffusion in the underdamped regime and show that it inherently violates the overdamped TUR for finite times. Based on numerical evidence, we then conjecture a bound for one-dimensional driven diffusion in a potential which is based on the result for free diffusion. This bound converges to the known overdamped TUR in the corresponding limit. Moreover, the conjectured bound holds for observables that involve higher powers of the velocity as long as the observable is odd under time-reversal. Finally, we address the applicability of this bound to underdamped dynamics in higher dimensions.