Relaxation Time of Quantized Toral Maps

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dc.creatorFannjiang, A.
dc.creatorNonnenmacher, S.
dc.creatorWolowski, L.
dc.date2004-06-24
dc.date2005-03-03
dc.date.accessioned2026-07-07T04:31:17Z
dc.date.available2026-07-07T04:31:17Z
dc.descriptionWe introduce the notion of the relaxation time for noisy quantum maps on the 2d-dimensional torus - a generalization of previously studied dissipation time. We show that relaxation time is sensitive to the chaotic behavior of the corresponding classical system if one simultaneously considers the semiclassical limit ($\hbar$ -> 0) together with the limit of small noise strength ($\ep$ -> 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime $\hbar<\ep^{E}$ << 1 (where E>1) in which classical and quantum relaxation times share the same asymptotics: in this regime, a quantized Anosov map relaxes to equilibrium fast, as the classical map does. As an intermediate result, we obtain rigorous estimates of the quantum-classical correspondence for noisy maps on the torus, up to times logarithmic in $\hbar^{-1}$. On the other hand, we show that in the ``quantum regime'' $\ep$ << $\hbar$ << 1, quantum and classical relaxation times behave very differently. In the special case of ergodic toral symplectomorphisms (generalized ``Arnold's cat'' maps), we obtain the exact asymptotics of the quantum relaxation time and precise the regime of correspondence between quantum and classical relaxations.
dc.descriptionLaTeX, 27 pages, former term dissipation time replaced by relaxation time, new introduction and references
dc.identifierhttps://arxiv.org/abs/math-ph/0406055
dc.identifierhttp://arxiv.org/abs/math-ph/0406055
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/57759
dc.subjectMathematical Physics
dc.subjectDynamical Systems
dc.subject81Q50, 37D20, 46L57
dc.titleRelaxation Time of Quantized Toral Maps
dc.typetext

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