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

In nonequilibrium systems, the relative fluctuation of a current has a universal tradeoff 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 breaking the microscopic time-reversal symmetry. 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.

Figure 1

A illustration describing the system.

Figure 2

A schematic description representing the relation between the original $\eta$ dynamics and the naive $\xi$ dynamics.

Figure 3

Duration-time-dependent behavior of the TUR factors (a) for work ${\mathcal{Q}}_W$, (b) for heat ${\mathcal{Q}}_Q$, and (c) for the entropy production ${\mathcal{Q}}_S$. The symbols represent the numerical data from the underdamped dynamics with $m_0=0.001$. Lines represent the analytic results obtained in the overdamped $\eta$ dynamics. Black dotted lines indicate the modified TUR bound in Eq. (45). We fixed (a) $B_0=0.5$ with several values of ${\epsilon }_0=0.1$ (red circle), 0.5 (blue triangle), and 1 (cyan square) for work, (b) $B_0=0.75$ with ${\epsilon }_0=-0.6$ (red circle), $-0.5$ (blue triangle), 0.75 (cyan square), and 2 (violet diamond) for heat, and (c) $B_0=0.6$ and ${\epsilon }_0=-0.1$ (red circle), 0.6 (blue triangle), 1.875 (cyan square), and 2 (violet diamond) for entropy production.

Figure 4

Finite-mass TUR factors for work ${\mathcal{Q}}_W$ versus dimensionless observation time $t_0$. The symbols represent the numerical data for $m_0=0.001$ (blue up-pointing triangle), 0.01 (orange down-pointing triangle), and 0.1 (green right-pointing triangle) with fixed $B_0=0.5$ and ${\epsilon }_0=0.4$. The horizontal dotted line indicates the modified TUR bound for the overdamped limit and the vertical dashed line indicates the $t_0=0.4$ line. The inset shows the plots for a wider time range, where the vertical dashed line indicates the $t_0=1.5$ line.