Python-Based Visualization of Fermi-Dirac and Bose-Einstein Distributions in Three-Dimensional Ideal Quantum Gases

Abstract

Quantum statistics describe the behavior of indistinguishable particles through the Fermi–Dirac (FD) and Bose–Einstein (BE) distributions. This study compares both distributions directly to emphasize their fundamental statistical differences. A Python-based computational model was developed to visualize the contrasting behaviors of FD and BE distributions in a unified 3D representation. The simulation uses NumPy and Matplotlib with an energy interval of 0.01–1.00, temperatures ranging from 100 K to 600 K, and a chemical potential of  The results show that the FD distribution forms a sharp profile near the Fermi energy and becomes step-like at low temperatures, while the BE distribution rises sharply as the chemical potential is approached, reflecting the characteristic mathematical divergence of Bose–Einstein statistics rather than indicating actual Bose–Einstein condensation. As temperature increases, the differences between the Fermi–Dirac and Bose–Einstein distributions become less pronounced, consistent with the expected high-temperature limit. The model yields an average relative deviation below 0.5%, confirming numerical stability and accuracy. This work provides an open-source Python framework for the visualization and comparative analysis of ideal quantum statistical distributions, intended primarily for educational and computational purposes rather than for predicting new physical phenomena.