The distribution of heat fluctuations in resistively-coupled dual temperature heat baths

A path integral approach used earlier to calculate the exact heat distribution function of a harmonically trapped Brownian oscillator at a fixed temperature (Chatterjee and Cherayil 2010 Phys. Rev. E 82 051104) is extended in this paper to a consideration of heat fluctuations in a dual temperature system. Our new calculations complement recent experimental data obtained by Ciliberto et al (2013 J. Stat. Mech. P12014) on the stochastic thermodynamics of an electrical circuit made up of coupled resistors maintained at two distinct temperatures. Measurements of various thermodynamic quantities in this system, including the heat, work and energy, reveal trends that represent interesting generalizations of results for the single temperature case. In particular, the measured distribution of the heat exchanged at one of the reservoirs is found to agree qualitatively with a fluctuation relation applicable at long times. In the present work, we exploit the formal equivalence between the electrical circuit and a system of coupled Brownian oscillators to derive, within the path integral formalism and for a special set of parameter values, an exact integral representation-valid for all times-of the total heat distribution function of the system. We find that the infinite-time limit of this distribution shows interesting departures from its expected behavior.