Regular and Irregular Motion in Classical And Quantum Systems

Abstract

It is now widely known that simple, nonlinear, deterministic dynamical systems can exhibit chaotic motion, which is unpredictable even in principle [1–4]. By simple, we mean that the system has only a few degrees of freedom N, say N = 2. By deterministic, we mean that the laws of motion are perfectly known locally; they are given by some velocity vector field in phase space. Time evolution of the system is governed by some ordinary differential equation, so that the future and the past are completely determined by the initial state of the system. The motion in time is obtained by finding the integral curves of the vector field (i.e., by solving the differential equation). This can always be done (at least numerically) for small times; and yet, the motion can still be unpredictable for large times. The reason for such a chaotic motion lies in the instability of trajectories, which can separate exponentially as time goes on. The motion thus displays a sensitive dependence on initial conditions. In such a case, the number of accurately calculated digits will decrease in proportion to the time elapsed from the initial condition, so that after a finite time our prediction will be completely wrong, no matter how good the measurement of the initial state was. An exponential instability can result in a type of motion which is as random as the outcome of tossing a coin. In fact, there is a whole hierarchy of chaotic systems, ranging from Bernoulli systems (most chaotic), through mixing and ergodic systems, down to almost-integrable (KAM) systems. An ergodic (or more chaotic) system will pass arbitrarily close to any point on the energy surface in phase space.