Comparison of Three Types of Statistical Distributions

1. Introduction

In statistical mechanics, particles in thermal equilibrium follow different distribution laws depending on their nature.

The three important statistical distributions are:

Applicable to classical distinguishable particles.

\[ f_{MB}(\epsilon) = e^{-\beta(\epsilon - \mu)} \]

Applicable to fermions (particles with half-integer spin).

\[ f_{FD}(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} + 1} \]

Applicable to bosons (particles with integer spin).

\[ f_{BE}(\epsilon) = \frac{1}{e^{\beta(\epsilon - \mu)} - 1} \]

Property	Maxwell–Boltzmann	Fermi–Dirac	Bose–Einstein
Particle Type	Classical particles	Fermions	Bosons
Spin	Any	Half-integer	Integer
Distribution Function	\( e^{-\beta(\epsilon-\mu)} \)	\( \frac{1}{e^{\beta(\epsilon-\mu)}+1} \)	\( \frac{1}{e^{\beta(\epsilon-\mu)}-1} \)
State Occupation	No restriction	Maximum 1 per state	Unlimited
Quantum Effect	Negligible	Important at low T	Important at low T
Low Temperature Behavior	Classical limit fails	Fermi energy dominates	Bose condensation possible

At high temperature and low density:

\[ e^{\beta(\epsilon - \mu)} \gg 1 \]

Both FD and BE distributions reduce to:

\[ f(\epsilon) \approx e^{-\beta(\epsilon - \mu)} \]

Thus, Maxwell–Boltzmann statistics is the classical limit of both Fermi–Dirac and Bose–Einstein statistics.

The fundamental difference between FD and BE statistics arises due to the symmetry properties of the wave function:

This leads to completely different macroscopic behaviors such as degeneracy pressure and Bose–Einstein condensation.