Rayleigh's Distribution, Wigner's Surmise and Equation of the Diffusion

Abstract

Non-Gaussian distributions occur in systems that do not follow strictly the prescriptions of standard statistics. Important example of non-Gaussian statistics is distribution which was introduced by Lord Rayleigh in connection with the problem of interference of harmonic oscillations with random phases [1, 2]. This distribution is known also as Wigner's surmise distribution giving a remarkably good description of the level repulsion observed in neutron scattering [3, 4]. As the mathematical form of both is identical, we call them Rayleigh Wigner's distribution (RWD). Rayleigh Wigner's distribution appears in computation of the large zeros of Riemann's zeta function on the critical line, which according to the Montgomery Odlyzko law have the same statistical properties as the distribution of eigenvalue spacings in a Gaussian unitary ensemble [5 9]. Rayleigh Wigner's distribution is also met in social sciences; e.g. the bus system in Cuernavaca, Mexico, is subject to this distribution [10]. This wide range of applications of RWD is a new illustration of Wigner's opinion on unreasonable e ectiveness of mathematics [11].