Quantum fluctuation relations

Abstract

We present here a set of lecture notes on exact fluctuation relations. We prove the Jarzynski equality and the Crooks fluctuation theorem, two paradigmatic examples of classical fluctuation relations. Finally we consider their quantum versions, and analyze analogies and differences with the classical case.Keywords: quantum thermodynamics; fluctuation relations. IntroductionWe present here the notes of three lectures given by one of us at the "Fifth International Workshop on Mathematical Foundations of Quantum Mechanics and its applications" held in February 2017 in Madrid at the Instituto de Ciencias Matemáticas (ICMAT).We will consider some results about fluctuation theorems both for classical and for quantum systems, a research topic that recently has attracted a great deal of attention. The statistical mechanics of classical and quantum systems driven far from equilibrium has witnessed quite recently a sudden development with the discovery of various exact fluctuation theorems which connect equilibrium thermodynamic quantities to non-equilibrium ones. There are excellent reviews on this topic, which cover both classical [1] and quantum fluctuation relations [2,3]. Here we will follow more closely the exposition by Campisi, Hänggi and Talkner [3], to which we refer the reader for further information.In the first lecture we will recall the derivation of Einstein's fluctuationdissipation relation for a Brownian particle, which is the inception of classical fluctuation relations. Moreover, we will identify the fundamental ingredients which are already present in this early derivation. Then, we will consider the GreenKubo formula, which represents the first general approach to quantum fluctuationdissipation relations. The link between the correlation function of a quantum system and its linear response function will be shown and the classical limit of the Green-Kubo formula will be considered.