Curvature distribution of chaotic quantum systems: Universality and nonuniversality

Abstract

The parametric motion of eigenvalues of two chaotic quantum systems, a stadium billiard and a kicked rotator, is studied to investigate the curvature distribution introduced by Gaspard etal. [Phys. Rev. A 42, 4015 (1990)]. By using an average of a statistical quantity over the parameter values along the motion, we obtain the curvature distribution and confirm the predicted universality of its tail behavior. We also show a nonuniversal characteristic at small curvatures, which is attributed to the existence of solitonlike structures.

A relation between billiard geometry and the temperature of its eigenvalue gas

“…A similar relation was already derived by Berry in his work on the spectral rigidity, but with a Lorentzian smearing of the deltafunctions [17]. The Gauss functions on the right-hand sides of equations (26) and (27) lead to a cut-off of the sums at lengths of the order of −1 . Thus in the limit → 0 the very long orbits give the dominating contributions to the sums on the right-hand sides of equations (26) and (27).…”

“…The agreement between theory and experiment is again good. We were not able to see deviations from equation (37) caused by the bouncing ball as they are reported, for example, for the stadium billiard [27,9] and the Sinai billiard [25]. These deviations should become manifest mainly in the region of small curvatures where the precision of our data was not sufficient to allow a reliable determination of the curvatures.…”

“…The Gauss functions on the right-hand sides of equations (26) and (27) lead to a cut-off of the sums at lengths of the order of −1 . Thus in the limit → 0 the very long orbits give the dominating contributions to the sums on the right-hand sides of equations (26) and (27). One can therefore replace l p andl p by N p l and N p l , respectively, where N p is the number of segments making up the orbit, and l is the average length of a segment.…”

“…As the step from equations (26) and (27) to equation (28) is crucial, two remarks may be appropriate [21]. First, it is not possible to argue from the exponential increase of the number of periodic orbits with length that the long orbits dominate the sums.…”

“…Equations (31) and (32) are used in the next section for the comparison of the experimental results with the theoretical predictions. The final results no longer contain any ingredients from periodic orbit theory because of the complete cancellation of the sums in equations (26) and (27). One may suspect that a more direct way to arrive at equation (32) exists without making use of periodic orbit theory.…”