# Entropic fluctuations in quantum statistical mechanics—an introduction

# Entropic fluctuations in quantum statistical mechanics—an introduction

This chapter presents a self-contained introduction to some recent developments in nonequilibrium quantum statistical mechanics. In the elementary framework of finite dimensional quantum systems it introduces the concept of entropy production and discusses various generating functionals which encode the statistical properties of its fluctuations. It explores the physical interpretations of these functionals and their mathematical properties. In particular, it investigates their relations to linear response theory, full counting statistics, and quantum hypothesis testing. The chapter discusses very briefly more technical issues linked to the thermodynamic limit as well as to the large time limit. These two limits turn the elementary, finite time fluctuation theory into a powerful machinery, which yields important information on the asymptotic behaviour of infinitely extended quantum systems, e.g., open systems coupled to infinite reservoirs. An essential mathematical tool to deal with such a system is Tomita–Takesaki's modular theory of von Neumann algebras. The power of this theory is somewhat shadowed by its technical aspects. Finite dimensional quantum systems are special since all the structures and results of this machinery can be described by elementary tools.

*Keywords:*
nonequilibrium quantum statistical mechanics, open quantum systems, entropy production, quantum friction, entropic fluctuations, fluctuation relations, trace inequalities, linear response theory, quantum transport, full counting statistics

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