On the Quantum Statistical Distributions Describing Finite Fermions and Bosons Systems

Elsayed K Elmaghraby

doi:10.4236/jmp.2011.211154

Abstract

A century old methodology for deriving statistical distribution using approximate Stirling’s formulation of the factorial becomes questionable. By avoiding the use of exaggerated approximations, a new picture of the energy distribution of fermions and bosons are presented. Energy distribution among fermions (or bosons) in systems with finite degeneracy are found to be degeneracy dependent. The presented point of view explains, successfully, presence of degeneracy pressure in ultra-cooled Fermi gas and predicts the minimum accessible temperature for finite degeneracy fermions system.

Highlights

Energy distribution among limited number of particles with finite degeneracy was a confusing issue when dealing with nuclear reaction/interaction
It is of the interest in pre- equilibrium reaction [1,2] in which low degeneracy states are occupied by finite number of excitons that exist together for very short time compared to the total reaction time; very long as compared to the nucleonnucleon interaction time
As a matter of quantum nature of physical system, non-degenerate and low degeneracy systems are considered finite; which means that systems from nuclei to nanoparticle are finite and their properties may be comparable

Summary

Introduction

Energy distribution among limited number of particles with finite degeneracy was a confusing issue when dealing with nuclear reaction/interaction. It is of the interest in pre- equilibrium reaction [1,2] in which low degeneracy states are occupied by finite number of excitons that exist together for very short time compared to the total reaction time; very long as compared to the nucleonnucleon interaction time. The current statistical description of the physical ensemble needs adjustment in order to follow proper justification of the definition of number and equivalence (or even non-equivalence) of a priori probabilities. More precise methodology is used to avoid usage of Stirling’s approximation that is used to derive the asymptotic MB, FD, and BE formulae

Dilemma of the Factorial

Maximum Entropy and Energy Distribution

Case of Fermions

Case of Bosons

Conclusions