New Type of Soliton Equation Described Some Statistical Distributions and Nonlinear Equations Unified Quantum Statistics

TL;DR

This paper introduces a novel soliton equation that models various statistical distributions, including quantum distributions, and demonstrates how nonlinear equations can unify different statistical behaviors.

Contribution

It presents a new soliton equation linking classical and quantum statistical distributions, extending to nonlinear Klein-Gordon and Dirac equations for unification.

Findings

Solutions describe Cauchy, normal, and Student distributions

Extension yields exponential and Fermi-Dirac distributions

Nonlinear equations unify Bose-Einstein and Fermi-Dirac distributions

Abstract

We proposed a new type of soliton equation, whose solutions may describe some statistical distributions, for example, Cauchy distribution, normal distribution and student distribution, etc. The equation possesses two characters. Further, from an extension of this type of equation we may obtain the exponential distribution, and the Fermi-Dirac distribution in quantum statistics. Moreover, by using the method of the soliton-solution, the nonlinear Klein-Gordon equation and nonlinear Dirac equations may derive Bose-Einstein and Fermi-Dirac distributions, respectively, and both distributions may be unified by the nonlinear equation.