Distribution of entropy production in a single-electron box

Recently, the fundamental laws of thermodynamics have been reconsidered for small systems. The discovery of the fluctuation relations(1-5) has spurred theoretical(6-13) and experimental(14-25) studies. The concept of entropy production has been extended to the microscopic level by considering stochastic trajectories of a system coupled to a heat bath. However, this has not been studied experimentally if there are multiple thermal baths present. Here, we measure, with high precision, the distributions of microscopic entropy production in a single-electron box consisting of two islands with a tunnel junction. The islands are coupled to separate heat baths at different temperatures, maintaining a steady thermal non-equilibrium. We demonstrate that stochastic entropy production(8,10-12,17,20,25,26) from trajectories of electronic transitions is related to thermodynamic entropy production from dissipated heat in the respective thermal baths. We verify experimentally that the fluctuation relations for both definitions are satisfied. Our results reveal the subtlety of irreversible entropy production in non-equilibrium.