Quantum Statistical Distributions (2D)

This 2D companion trades the original's decorative sphere-and-ring scene for the actual quantum-statistical mechanics the title promises: a live plot of the real occupation-number formulas — Fermi-Dirac n(E) = 1/(e^((E-μ)/kT)+1), Bose-Einstein n(E) = 1/(e^((E-μ)/kT)-1) and the classical Maxwell-Boltzmann limit n(E) = e^(-(E-μ)/kT) — evaluated against a free-particle density of states g(E) ∝ √E, with the chemical potential μ and temperature kT as draggable parameters. Total particle number N and average energy ⟨E⟩ are computed as running numeric integrals over the plotted curve rather than displayed as fixed percentages, and a column of energy-level dots below the plot fills up level by level exactly as the occupation formula predicts — thinning toward a sharp step under Fermi-Dirac statistics, or piling up near the ground state as μ approaches 0 under Bose-Einstein statistics.