Wigner spacing distribution in classical mechanics

Abstract

The Wigner spacing distribution has a long and illustrious history in nuclear physics and in the quantum mechanics of classically chaotic systems. In this paper, a long-overlooked connection between the Wigner distribution and \emph{classical} chaos in two-degree-of-freedom (2D) conservative systems is introduced. In the specific context of fully chaotic 2D systems, the hypothesis that typical pseudotrajectories of a canonical Poincar\'{e} map have a Wignerian nearest-neighbor spacing distribution (NNSD), is put forward and tested. Employing the 2D circular stadium billiard as a generic test case, the NNSD of a typical pseudotrajectory of the Birkhoff map is shown to be in excellent agreement with the Wigner distribution. The relevance of the higher-order Wigner surmises from random matrix theory are also illustrated.