Thermodynamic uncertainty relation in the overdamped limit with a magnetic Lorentz force

TL;DR

This paper derives a modified thermodynamic uncertainty relation for overdamped Langevin systems with magnetic Lorentz forces, revealing conditions under which the conventional TUR can be violated due to broken time reversal symmetry.

Contribution

It analytically extends the TUR to include magnetic Lorentz forces in overdamped systems, showing how the relation is scaled and can be violated.

Findings

Modified TUR scaled by effective temperature ratio

Violation of TUR in presence of Lorentz force

Analytical and numerical verification in a solvable model

Abstract

In nonequilibrium systems, the relative fluctuation of a current has a universal trade-off relation with the entropy production, called the thermodynamic uncertainty relation (TUR). For systems with broken time reversal symmetry, its violation has been reported in specific models or in the linear response regime. Here, we derive a modified version of the TUR analytically in the overdamped limit for general Langevin dynamics with a magnetic Lorentz force causing time reversal broken. Remarkably, this modified version is simply given by the conventional TUR scaled by the ratio of the reduced effective temperature of the overdamped motion to the reservoir temperature, permitting a violation of the conventional TUR. Without the Lorentz force, this ratio becomes unity and the conventional TUR is restored. We verify our results both analytically and numerically in a specific solvable system.