PHYSICAL REVIEW E VOLUME 51, NUMBER 6 JUNE 1995 Anomalous diffusion and the correspondence principle Roberto Roncaglia, Luca Bonci, Bruce J. West, and Paolo Grigolini ' ' Dipartimento di Eisica dell Uninersita di Pisa, Piazza Torricelli 2, 56200 Pisa, Italy Center for Nonlinear Science, University of North Tezas, P. O. Boz 5868, Denton, Tezas 76208 Istituto di Biofisica del Consiglio Nazionale detle Ricerche, via San Lorenzo 28, 56187 Pisa, Italy (Received 22 July 1994) We study the quantum behavior of the standard map in the so-called accelerator state, which, within the theoretical framework of classical physics, would result in anomalous diffusion. In agree- ment with the behavior of the systems, which classically exhibit full chaos and are consequently characterized by positive Lyapunov coeKcients A, quantum uncertainty increases very quickly and leads to the breakdown of the overwhelming majority of the classical trajectories in a time tz — — (1/A) In(1/h). In the case of normal diff'usion, the diffusion process is unaffected by this rapid transition from classical to quantum physics. However, in the case of anomalous diffusion, we find the existence of a new breakdown process, corresponding to a statistical departure of quantum from classical dynamics. We argue that this new kind of breakdown, which does not have anything to do with the well known phenomenon of localization, takes place on a time scale te = (1/Af) In(1/h) larger than t„and with Lyapunov coeKcient Ay determined essentially by the stochastic trajectories moving on the border between the stochastic sea and the accelerator islands. If our arguments are confirmed, they would lead to the possibility of observing the breakdown of the correspondence principle in the statistical sense in times compatible with experimental observation. PACS number(s): 05. 45.+b, 03. 65.Bz I. INTRODUCTION One of the most striking aspects of the dynamics of quantum systems that would be classically chaotic is the discovery of possible new channels for the macroscopic manifestation of quantum mechanics. The majority of authors [I — 4], with but a few exceptions [5], agree on this issue with different arguments leading, however, to equivalent conclusions. We think that probably the most intuitive way to reach this conclusion is given by the picture recently used by Zurek and Paz [4]. Following them [4], let us consider a quantum system in the phys- ical condition where, according to a traditional wisdom, quantum dynamics is supposed to be virtually coincident with classical mechanics. Let us consider, for instance, a driven one-dimensional system, namely, a particle mov- ing in a time-dependent potential V(q, t). Even though we provide a direct numerical integration of the quantum dynamics in the text, in this section, for pedagogical rea- sons, we discuss the qualitative nature of the quantum system using a phase-space equation of evolution. We adopt the Wigner formalism [6] leading to the equation of motion for the quasiprobability pgr(q, p; t), 8 BtP (q p't) = [L -+ Lct]P (q p't) where L ~ „coincides with the classical Poisson bracket and I g is the quantum contribution reading (2) Let us consider the case where the Planck constant 5 is pl, (q, p;t) = exp(L. ) „t)pL, (q, p;0) . The quantum evolution of the system is given in terms of the Wigner density p~(q, p; t) = exp[(L, ~ „+ Lg)t]pl, (q, p; 0) . (4) The validity of the classical picture is broken when p~(q, p; t) significantly departs &om pL, (q, p; t). In the case of ordinary classical motion the time required for this breakdown to take place is inversely proportional to the Planck constant to a power of the order of unity and consequently a time so astronomically large as to result in a complete fulfillment of the expectations of the cor- respondence principle [I]. In the case where the classical motion is fully chaotic a completely different situation occurs. The Liouville den- sity undergoes a rapid process of fragmentation and the originally smooth distribution quickly develops whorls and tendrils and becomes more and more finely frag- mented with increasing time. This process of increasing fragmentation has the efFect of enhancing the role of the operator Lg due to the sharp gradients in the fragmented distribution. This operator would rigorously vanish in the case of linear systems, thereby making unlimited the time of validity of the correspondence principle. Even extremely small compared to the typical macroscopic ac- tion LI, namely, the volume of the phase space available to the system within the classical representation. Under this condition, where the classical picture is expected to hold, we adopt an initial distribution given by a smooth Liouville density pL, (q, p; 0). The classical evolution of the system is given in terms of the Liouville density 1063-651X/95/51(6)/5524(11)/$06. 00 51 5524 1995 The American Physical Society