On the validity of classical partition function

TL;DR

This paper presents a pedagogical computer-based experiment to illustrate the differences in partition functions for distinguishable, Bosonic, and Fermionic particles, highlighting the importance of indistinguishability and quantum statistics.

Contribution

It introduces an educational tool that demonstrates key differences in classical and quantum partition functions without relying on the single-particle partition function.

Findings

Clarifies the differences in partition functions for various particle types

Highlights the importance of indistinguishability in statistical mechanics

Provides an accessible teaching method for complex quantum concepts

Abstract

The partition function for a system of non-interacting $N-$particles can be found by summing over all the states of the system. The classical partition function for an ideal gas differs from Bosonic or Fermionic partition function in the classical regime. Students find it difficult to follow the differences arising out of incorrect counting by the classical partition function by missing out on the indistinguishability of particles and Fermi-Bose statistics. We present a pedagogical computer-based experiment to probe and demonstrate the key differences in the partition functions (i) $Q$, for the system of distinguishable and indistinguishable particles (ii) $B$, for Bosons and (ii) $F$, for Fermions without formally using the single-particle partition function.