Control of Fluctuations and Heavy Tails for Heat Variation in the Two-Time Measurement Framework

Abstract

We study heat fluctuations in the two-time measurement framework. For bounded perturbations, we give sufficient ultraviolet regularity conditions on the perturbation for the moments of the heat variation to be uniformly bounded in time, and for the Fourier transform of the heat variation distribution to be analytic and uniformly bounded in time in a complex neighborhood of 0.On a set of canonical examples, with bounded and unbounded perturbations, we show that our ultraviolet conditions are essentially necessary. If the form factor of the perturbation does not meet our assumptions, the heat variation distribution exhibits heavy tails. The tails can be as heavy as preventing the existence of a fourth moment of the heat variation.

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

“…Although obviously a stronger condition than the boundedness of V which was assumed to derive (1.1), this ultraviolet regularity condition holds in many models of physical interest; see Section 4. Moreover, the results of [BPR18] show that our regularity condition is essentially necessary for large fluctuations of total heat variation to be exponentially suppressed.…”

“…In this framework, while the derivation of the fluctuation relation becomes trivial, the equality of heat variation and work holds only for their first two moments (see Section 3.5 of [JOPP12]). In a related paper [BPR18], examples are provided where total heat variation distribution has infinite higher moments even if the interaction is bounded and its norm arbitrary small. We refer the reader to the introduction of the related paper [BPR18] for a detailed discussion of this issue.…”

“…In a related paper [BPR18], examples are provided where total heat variation distribution has infinite higher moments even if the interaction is bounded and its norm arbitrary small. We refer the reader to the introduction of the related paper [BPR18] for a detailed discussion of this issue.…”

“…These properties can be interpreted as statistical refinements of the heat conservation and second law of thermodynamics. As the results in the aforementioned paper [BPR18] show, this analysis is not a straightforward extension of the classical case, even when the interaction is bounded: ultraviolet regularity assumptions play an important role. We consider a system made of several baths, initially uncoupled, and then interacting, possibly but not necessarily by coupling them to a small system.…”

“…is uniformly bounded in a complex neighborhood of the origin and in a complex neighborhood of the couple of inverse temperatures β (our corresponding assumption is B β (α 0 , θ 0 ), precisely defined in Section 3). We call this assumption an ultraviolet regularity as in paradigmatic models of nonequilibrium statistical mechanics the condition translates directly to an ultraviolet regularity condition (see Section 4.2 and [BPR18]). Although obviously a stronger condition than the boundedness of V which was assumed to derive (1.1), this ultraviolet regularity condition holds in many models of physical interest; see Section 4.…”