2004/06/24 by A. Fannjiang, Fannjiang, A., S. Nonnenmacher +3
Mathematics · Physics and Astronomy · #37D20 #46L57 #81Q50 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.DS #math.MP #msc:37D20 #msc:46L57 #msc:81Q50
paper · pdf · doi:10.48550/arxiv.math-ph/0406055
LaTeX, 27 pages, former term dissipation time replaced by relaxation time, new introduction and references
arxiv created 2005/03/03 · arxiv updated 2009/12/01
We 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 (ℏ -> 0) together with the limit of small noise strength (\ep -> 0). Focusing on quantized smooth Anosov maps, we exhibit a semiclassical regime ℏ<\epE << 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 ℏ-1. On the other hand, we show that in the ``quantum regime'' \ep << ℏ << 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.