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arXiv:2002.10989 · Journal of Statistical Physics

Linear Response Theory and Entropic Fluctuations in Repeated Interaction Quantum Systems

Jean-François Bougron, Laurent Bruneau

Published Sep 25, 2020• 10 citations
arXivHTMLPDF
DOI
10.1007/s10955-020-02640-x

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We study a class of dynamical semigroups [Formula: see text] that emerge, by a Feynman–Kac type formalism, from a random quantum dynamical system [Formula: see text] driven by a Markov chain [Formula: see text]. We show that the almost sure large time behavior of the system can be extracted from the large [Formula: see text] asymptotics of the semigroup, which is in turn directly related to the spectral properties of the generator [Formula: see text]. As a physical application, we consider the case where the [Formula: see text]’s are the reduced dynamical maps describing the repeated interactions of a system [Formula: see text] with thermal probes [Formula: see text]. We study the full statistics of the entropy in this system and derive a fluctuation theorem for the heat exchanges and the associated linear response formulas.
Scattering of quantum fluctuations by a linear system is entirely characterized by impedance functions that also allow the determination of the system fluctuations. We present a generalization of this linear input-output theory to the case of a nonlinear scattering system. We define response functions that are similar to the susceptibility functions of linear response theory, but behave as noncommuting quantities. These functions contain a ``dynamical'' part determined by the relaxation of the system and a ``structural'' one related to the commutators between the system observables. The generalized linear input-output theory provides us with a complete description of the system fluctuations as well as of the output reservoir fields. The consistency of the results is ensured by general relations existing between the response functions and the correlation functions.
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.