Integral fluctuation theorem and generalized Clausius inequality for microcanonical and pure states

Fluctuation theorems are cornerstones of modern statistical mechanics and their standard derivations routinely rely on the crucial assumption of a canonical equilibrium state. Yet rigorous derivations of certain fluctuation theorems for microcanonical states and pure energy eigenstates in isolated quantum systems are still lacking and constitute a major challenge to theory. In this work we tackle this challenge and present such a derivation of an integral fluctuation theorem (IFT) by invoking two central and physically natural conditions, i.e., the so-called "stiffness" and "smoothness" of transition probabilities. Our analytical arguments are additionally substantiated by numerical simulations for archetypal many-body quantum systems, including integrable as well as nonintegrable models of interacting spins and hard-core bosons on a lattice. These simulations strongly suggest that "stiffness" and "smoothness" are indeed of vital importance for the validity of the IFT for microcanonical and pure states. Our work contrasts with recent approaches to the IFT based on Lieb-Robinson speeds and the eigenstate thermalization hypothesis.