Crossover from Diffusive to Ballistic Transport in Periodic Quantum Maps

We derive an expression for the mean square displacement of a particle whose motion is governed by a uniform, periodic, quantum multi-baker map. The expression is a function of both time, $t$, and Planck's constant, $\hbar$, and allows a study of both the long time, $t\to\infty$, and semi-classical, $\hbar\to 0$, limits taken in either order. We evaluate the expression using random matrix theory as well as numerically, and observe good agreement between both sets of results. The long time limit shows that particle transport is generically ballistic, for any fixed value of Planck's constant. However, for fixed times, the semi-classical limit leads to diffusion. The mean square displacement for non-zero Planck's constant, and finite time, exhibits a crossover from diffusive to ballistic motion, with crossover time on the order of the inverse of Planck's constant. We argue, that these results are generic for a large class of 1D quantum random walks, similar to the quantum multi-baker, and that a sufficient condition for diffusion in the semi-classical limit is classically chaotic dynamics in each cell. Some connections between our work and the other literature on quantum random walks are discussed. These walks are of some interest in the theory of quantum computation.Comment: Final version to appear in Physica D, Proceedings of the International Workshop and Seminar on Microscopic Chaos and Transport in Many-Particle Systems, Dresden, 2002; corrected a minor error in section 3.1, new section 4.

Imprints of log-periodic self-similarity in the stock market

Detailed analysis of the log-periodic structures as precursors of the financial crashes is presented. The study is mainly based on the German Stock Index (DAX) variation over the 1998 period which includes both, a spectacular boom and a large decline, in magnitude only comparable to the so-called Black Monday of October 1987. The present example provides further arguments in favour of a discrete scale-invariance governing the dynamics of the stock market. A related clear log-periodic structure prior to the crash and consistent with its onset extends over the period of a few months. Furthermore, on smaller time-scales the data seems to indicate the appearance of analogous log-periodic oscillations as precursors of the smaller, intermediate decreases. Even the frequencies of such oscillations are similar on various levels of resolution. The related value $\lambda \approx 2$ of preferred scaling ratios is amazingly consistent with those found for a wide variety of other complex systems. Similar analysis of the major American indices between September 1998 and February 1999 also provides some evidence supporting this concept but, at the same time, illustrates a possible splitting of the dynamics that a large market may experience.Comment: 13 pages, LaTeX-REVTeX, 4 PS figures. Significantly extended version to appear in The European Physical Journal