Non-Gaussian distributions occur in systems that do not follow the prescriptions of standard statistics. Prominent example of non-Gaussian statistics is Wigner’s surmise distribution giving a remarkably good description of the level repulsion observed in neutron scattering. 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. Wigner's distribution is also found in social sciences, e.g. the bus system in Cuernavaca, Mexico, is subject to this distribution.

Most non equilibrium systems do not have analytical solutions for the distribution and correlation functions. In this contribution the evolution of Wigner’s statistics is studied and a diffusion type equation with source term is proposed for this aim. The solution of this equation is the Wigner distribution of the type f(x,t) = a(t) x exp[-b(t) x^2], where a = a(t) and b = b(t) are functions of time t only, and x is the space variable.

1. Eugene P. Wigner, On the statistical distribution of the widths and spacings of nuclear resonance levels, Mathematical Proceedings of the Cambridge Philosophical Society Vol. 47, Iss. 04, 790-798 (1951).

2. H.L. Montgomery, The pair correlation of zeros of the Riemann zeta-function on the critical line, Proc. Symp. Pure Math. Providence 24, 181-193 (1973).

3. A.M. Odlyzko, The 10^{20}th zero of the Riemann zeta-function and 70 million of its neighbors, in ’Dynamical, spectral, and arithmetic zeta functions’ (San Antonio, TX, 1999), 139–144, Contemp. Math. 290, Amer. Math. Soc., Providence 2001.

4. N.M. Katz, P. Sarnak, Zeros of the zeta function and symmetry, Bulletin of The AMS Vol. 36, Number 1, January 1999, Pages 1-26.

5. Jinho Baik, Alexei Borodin, Percy Deift, Toufic Suidan, A model for the bus system in Cuernavaca (Mexico), Journal of Physics A: Mathematical and General, Vol. 39, Iss. 28, pp. 8965-8975 (2006).

6. Milan Krbálek and Petr Šeba, Spectral rigidity of vehicular streams (random matrix theory approach), J. Phys. A: Math. Theor. 42 (2009) 345001 (10pp).

• Legal notice:
  

  Copyright (c) Pielaszek Research, all rights reserved.
  The above materials, including auxiliary resources, are subject to Publisher's copyright and the Author(s) intellectual rights. Without limiting Author(s) rights under respective Copyright Transfer Agreement, no part of the above documents may be reproduced without the express written permission of Pielaszek Research, the Publisher. Express permission from the Author(s) is required to use the above materials for academic purposes, such as lectures or scientific presentations.
  In every case, proper references including Author(s) name(s) and URL of this webpage: https://science24.com/paper/25332 must be provided.

1. Student's t-distribution versus Zeldovich-Kompaneets solution of diffusion problem
2. Riemann zeta in biological and social events
3. The average behaviour of financial market by 2 scale homogenisation
4. From Riemann zeta through L-functions, random matrices, quantum chaos, brownian diffusion, critical collective phenomena ... to financial correlations