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We study the statistical properties of currents in two particular systems ofcapacitively coupled parallel transport channels. In the first system, eachtransport channel contains a single quantum dot in contact with two electronreservoirs. The second system we study is constituted of a double quantum dotcoupled to two electrodes and probed by a quantum point contact detector.Thermodynamic forces are applied to each transport channel that generatefluctuating stationary currents. The full counting statistics of the currentsis obtained starting from a microscopic Hamiltonian describing the electrondynamics. We verify that the joined probability distribution of the currents ineach channel satisfies a fluctuation theorem in the long-time limit. The issueof single-current fluctuation theorems for the marginal distribution of thecurrents in one of the two channels is also investigated. We show that in thelimit of large current ratio between both channels, a single-currentfluctuation theorem is satisfied individually for the slower circuit inagreement with experimental observations. This theorem involves an effectiveaffinity which depends on the thermodynamic forces applied to both channels andthe specific features of the system considered. A detailed study of theeffective affinity is made for the two aforementioned systems. Besides, weintroduce a criteria on the initial condition of the transport channels for theobservation of a fluctuation theorem at any time. This criteria is extended tothe case of single-current fluctuation theorems. Finally, we perform thenonequilibrium thermodynamic analysis of a double quantum dot probed by aquantum point contact in the presence of temperature and chemical potentialdifferences between the electrodes. A thermal machine is studied and shown toreach highest efficiencies at maximum power by fine tuning the double quantumdot spectrum.

Fluctuation theorem for quantum electron transport in mesoscopic circuits