Thermodynamic uncertainty relations for multi-terminal systems with broken time-reversal symmetry

We investigate the thermodynamic uncertainty relations (TURs) in steady-state transport for a multi-terminal system consisting of two conducting terminals and N-2 probe terminals, within the linear response regime under broken time-reversal symmetry. We independently derive exact bounds on the TURs for the steady-state particle and heat currents under a strong constraint on the Onsager coefficients. Based on our proposed exact bounds, the analysis reveals that the bounds differ for particle and heat currents and are dependent on the system parameters. Furthermore, we demonstrate that under specific parameter conditions, the TURs of the particle and heat currents have a unified minimum value that depends solely on the number of terminals.