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Quantum Dynamical Systems$
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Robert Alicki and Mark Fannes

Print publication date: 2001

Print ISBN-13: 9780198504009

Published to Oxford Scholarship Online: February 2010

DOI: 10.1093/acprof:oso/9780198504009.001.0001

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PRINTED FROM OXFORD SCHOLARSHIP ONLINE (oxford.universitypressscholarship.com). (c) Copyright Oxford University Press, 2021. All Rights Reserved. An individual user may print out a PDF of a single chapter of a monograph in OSO for personal use. date: 27 October 2021

Ergodic Theory

Ergodic Theory

(p.122) 7 Ergodic Theory
Quantum Dynamical Systems

Robert Alicki

Mark Fannes

Oxford University Press

This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and GNS-representation for quantum systems. It presents mixing and asymptotic Abelianness. It then discusses a number of examples with non-trivial algebraic structures: quasi-free Fermionic automorphisms, highly anti-commutative systems, and Powers–Price shifts. It is shown that under certain ergodic assumptions the fluctuations around ergodic means can be modelled by Bose fields in quasi-free states (Gaussian distributions), the other extreme cases lead to a free probability scheme with semi-circular distributions. Expanding maps and Lyapunov exponents for classical dynamics are briefly discussed and a possible quantum analog, horocyclic actions, is presented for a quantum cat map.

Keywords:   ergodicity, mixing, asymptotic Abelianness, quasi-free Fermionic automorphism, anti-commutative system, Powers-Price shift, quantum central limit theorem, free probability, Lyapunov exponent, expanding map

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