Quantum Statistics – The Bose–Einstein Distribution

Charles A. Ginsburg

Ta-Pei Cheng

We discuss another of Einstein’s statistical contributions to quantum theory: his work on quantum statistics. It was prompted by a paper that S. Bose sent to him in 1924 when the physics community had finally accepted the photon idea upon the discovery and analysis of Compton scattering. We present in detail Bose’s derivation of Planck’s distribution, using a particle approach. Einstein extended Bose’s analysis of radiation to systems of matter particles. Here he made the discovery of the astounding phenomenon of Bose–Einstein condensation. The ultimate understanding of Planck’s spectral distribution came about in modern quantum mechanics with its notion of indistinguishable particles, and the connection between spin and statistics. We also provide a modern discussion of Bose–Einstein condensation.

Murnia, Rania Dwicahyani, Neysya Ditha Ramadhani +2 more

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.

Lester Beltran

We investigate the hypothesis that within a combination of a 'number concept' plus a 'substantive concept', such as 'eleven animals', the identity and indistinguishability present on the level of the concepts, i.e., all eleven animals are identical and indistinguishable, gives rise to a statistical structure of the Bose-Einstein type similar to how Bose-Einstein statistics is present for identical and indistinguishable quantum particles.We proceed by identifying evidence for this hypothesis by extracting the statistical data from the World-Wide-Web utilizing the Google Search tool.By using the Kullback-Leibler divergence method, we then compare the obtained distribution with the Maxwell-Boltzmann as well as with the Bose-Einstein distributions and show that the Bose-Einstein's provides a better fit as compared to the Maxwell-Boltzmann's.