Possible Size Dependence of Distribution Functions of Classic, Boson, and Fermion Assemblies

TL;DR

This paper derives the size dependence of distribution functions for classical, bosonic, and fermionic systems without Stirling approximation, highlighting the role of Fermi energy or chemical potential, and aligns with some experimental observations on nanomaterials.

Contribution

It introduces a size-dependent distribution function derivation that avoids Stirling approximation, emphasizing the role of Fermi energy or chemical potential.

Findings

Distribution functions depend on particle size via Fermi energy or chemical potential.

Results match some experimental reports on nanometer-sized materials.

Deviation from traditional models at small particle numbers is addressed.

Abstract

I derived the size dependence of distribution function for classic, boson, and fermion assemblies. I did not use the Stirling approximation so that deviation contributed by this approximation at small number of particles can be avoided. I identified that the size dependence of the distribution function is contained in the fermi energy or chemical potential. My results seem to match few reports on the dependence of fermi energy or chemical potential on particle size of several nanometer sized materials