Heat and charge current fluctuations and the time dependent coefficient of performance for a nanoscale refrigerator

We theoretically investigate the coefficient of performance (COP) of a mesoscopic thermoelectric refrigerator realized by using a tunnel junction. We analyze the influence of particle and heat current fluctuations on the COP out of the equilibrium regime. We calculate the average COP by using full counting statistics and find that it depends on the measurement time $\tau$. The deviation from the macroscopic COP value can be expressed with the Skellam distribution at all times. This result enables us to improve the Gaussian approximation valid within the linear response regime, which cannot predict the average COP in the limit of $\tau \to 0$. We illustrate the time dependence of the average COP and find that in the short-time regime, the average COP possesses a minimum. In order to confirm the physical consistency far from equilibrium, we propose checking the correlation coefficient between the particle and the heat currents in addition to the positivity of the entropy production rate.