Physica E 9 (2001) 436–442 www.elsevier.nl/locate/physe Exotic stochastic processes from quantum chaotic environments Dimitri Kusnezov a ; 1 , Aurel Bulgac b; ∗ , Gui Do Dang c a Center for Theoretical Physics, Sloane Physics Laboratory, Yale University, New Haven, CT 06520-8120, USA b Department of Physics, University of Washington, P.O. Box 351560, Seattle, WA 98195-1560, USA c Laboratoire de Physique Th eorique, Universit e de Paris-Sud Bˆ atiment 211, 91405, Orsay, France Abstract Stochastic processes are shown to emerge from the time evolution of complex quantum systems. Using parametric, banded random matrix ensembles to describe a quantum chaotic environment, we show that the dynamical evolution of a particle coupled to such environments displays a variety of stochastic behaviors, ranging from turbulent diusion to L evy processes and Brownian motion. Dissipation and diusion emerge naturally in the stochastic interpretation of the dynamics. This approach provides a derivation of a fractional kinetic theory in the classical limit and leads to classical L evy dynamics. ? 2001 Elsevier Science B.V. All rights reserved. PACS: 05.45.Mt; 03.65.Sq; 05.30.-d; 05.40.Fb Keywords: L evy ights; Random matrix theory; Quantum dissipation; Kinetic equations 1. Introduction The understanding of how stochastic processes emerge from classical dynamical systems is closely related to classical chaos. Often one nds that the dy- namics is non-Gaussian, displaying either enhanced or dispersive behavior [1]. One can nd such nd such behavior in the interaction of slow and fast de- grees of freedom in many-body systems [2], in tracer * Corresponding author. Fax: +1-206-685-9829. E-mail address: bulgac@phys.washiington.edu (A. Bulgac). 1 The present work has been partially supported by DOE. The Laboratoire de Physique Th eorique is a Unit e Mixte de Recherche du C.N.R.S., UMR 8627. diusion in turbulent backgrounds such as the atmo- sphere, or random potentials, and many more [3]. The wide variety of processes which exhibit anomalous behavior in the transport has led to a variety of the- oretical eorts, including fractional extensions of ki- netic theory [4 – 6], random walks in random potentials [7,8], power law noise in generalized Langevin equa- tions [9], stochastic webs [4,5] and L evy walks and ights [3,10,11]. While the common thread to these approaches are generalizations of Brownian motion known as L evy stable laws (discussed below), there is no common theoretical foundation. For instance, frac- tional kinetic equations are postulated in such a man- ner as to provide the desired diusion through scaling arguments. 1386-9477/01/$ - see front matter ? 2001 Elsevier Science B.V. All rights reserved. PII:S1386-9477(00)00241-1