Heat Conservation and Fluctuations Between Quantum Reservoirs in the Two-Time Measurement Picture

This work concerns the statistics of the Two-Time Measurement definition of heat variation in each reservoir of a thermodynamic quantum system. We study the cumulant generating function of the heat flows in the thermodynamic and large-time limits. It is well-known that, if the system is time-reversal invariant, this cumulant generating function satisfies the celebrated Evans-Searles symmetry. We show in addition that, under appropriate ultraviolet regularity assumptions on the local interaction between the reservoirs, it satisfies a translation-invariance property, as proposed in Andrieux et al. (in New J Phys 11(4):043014, 2009). We particularly fix some proofs of the latter article where the ultraviolet condition was not mentioned. We detail how these two symmetries lead respectively to fluctuation relations and a statistical refinement of heat conservation for isolated thermodynamic quantum systems. As in Andrieux et al. (in New J Phys 11(4):043014, 2009), we recover the fluctuation-dissipation theorem in the linear response theory, short of Green-Kubo relations. We illustrate the general theory on a number of canonical models.