Nearest-neighbor level spacing distributions: On the transition from the regular to the chaotic regimes

The statistical mechanics of the distribution of Hamiltonian matrix elements in the intermediate coupling regime is discussed. Starting from an integrable limit, a coupling of increasing strength is imposed in a manner pioneered by Dyson [J. Math. Phys. 3, 421 (1962)]. In the long-time limit one approaches the Gaussian orthogonal ensemble distribution. Equivalent results can be obtained by the method of maximum entropy. The special case of a two-level system is worked out in detail. Except for the strictly integrable limit, the spacing distribution for small spacings is shown to be proportional to the spacing. The intensity distribution for small intensities is proportional to the Porter-Thomas distribution. The dependence of these distributions on the coupling-strength parameter, which measures the deviations from regular behavior, is discussed.