Statistical Theory of the Energy Levels of Complex Systems. II

The distribution function of spacings S between nearest neighbors, in a long series of energy levels with average spacing D, is studied. The statistical properties of S are defined in terms of an ensemble of systems described in a previous paper. For large values of t = (πS/2D), it is shown that the distribution of S can be deduced from the thermodynamical properties of a certain model. The model, which replaces the eigenvalue distribution by a continuous fluid, can be studied by the methods of classical electrostatics, potential theory, and thermodynamics. In this way the distribution function of spacings S is found to be asymptotically Q(t)=At17/8 exp[−14t2−12t]for large t. The numerical constant A can in principle not be determined from such a continuum model. Reasons are given for considering the remaining factors in the formula for Q(t) to be reliable.