Nonlinear Sciences – Chaotic Dynamics
Scientific paper
2005-03-23
Phys. Rev. E 72, 046207 (2005)
Nonlinear Sciences
Chaotic Dynamics
31 pages, 17 figures
Scientific paper
10.1103/PhysRevE.72.046207
We argue semiclassically, on the basis of Gutzwiller's periodic-orbit theory, that full classical chaos is paralleled by quantum energy spectra with universal spectral statistics, in agreement with random-matrix theory. For dynamics from all three Wigner-Dyson symmetry classes, we calculate the small-time spectral form factor $K(\tau)$ as power series in the time $\tau$. Each term $\tau^n$ of that series is provided by specific families of pairs of periodic orbits. The contributing pairs are classified in terms of close self-encounters in phase space. The frequency of occurrence of self-encounters is calculated by invoking ergodicity. Combinatorial rules for building pairs involve non-trivial properties of permutations. We show our series to be equivalent to perturbative implementations of the non-linear sigma models for the Wigner-Dyson ensembles of random matrices and for disordered systems; our families of orbit pairs are one-to-one with Feynman diagrams known from the sigma model.
Altland Alexander
Braun Petr
Haake Fritz
Heusler Stefan
Müller Sebastian
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