Heat conservation and fluctuations between quantum reservoirs in the Two-Time Measurement picture

TL;DR

This paper analyzes the statistical properties of heat exchange in quantum systems using the Two-Time Measurement approach, revealing symmetries and fluctuation relations under certain regularity conditions.

Contribution

It establishes translation-invariance of the cumulant generating function under ultraviolet regularity assumptions, fixing gaps in previous proofs and deriving fluctuation relations and heat conservation insights.

Findings

Proves translation-invariance of the cumulant generating function.

Derives fluctuation relations from symmetries in heat statistics.

Shows heat conservation refinement in quantum thermodynamic systems.

Abstract

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. New J. Phys. 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 \emph{et al.} New J. Phys. 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.