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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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Finite Quantum Systems

Finite Quantum Systems

(p.225) 12 Finite Quantum Systems
Quantum Dynamical Systems

Robert Alicki

Mark Fannes

Oxford University Press

Quantum systems with a discrete dynamical spectrum, such as finite dimensional systems or particles confined to a finite volume, cannot exhibit true ergodic behaviour. Nevertheless, scaling limits can be considered. This chapter focuses on the behaviour of entropy for small value of Planck's constant, i.e. how the classical limit is reached. It also considers other classical limits such as the limit of large angular momentum for the kicked top. Numerical computations are needed to compute the actual value of the total entropy produced up to a finite time, but its saturation behaviour for long times can be established analytically. The operational partitions that eventually lead to the largest entropy can also be determined. As a complementary tool for the analysis of randomizing behaviour, the chapter sketches the use of the Gram matrix to study the statistics of return times.

Keywords:   finite system, kicked top, scaling limit, Gram matrix, return time

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