Relaxation to equilibrium of quantized chaotic states

Abstract

Classical chaotic systems proceed to equilibrium by a mixing process in the phase space. We study this relaxation to equilibrium for quantized, bound chaotic maps on the torus. The Ruelle-Pollicott resonances that describe the mixing of classical chaotic system has been derived for the lazy-baker map, which is a non-uniformly hyperbolic system. The corresponding issue of quantifying mixing in Hilbert space has been discussed in this thesis and a correspondence between the classical and quantum relaxation has been shown for the same map. Later the e ect of a random unitary transformation on the quantum relaxation has been shown to be an immediate relaxation to the equilibrium. An average of the quantum relaxation over all such random unitary transformations has been analytically performed. The average of the relaxation uctuations about the equilibrium is derived to have a simple structure under certain conditions, re ecting the universal properties of the concerned random matrix ensemble.