Quantum Gas Statistics Visualizer

An ideal gas of $N$ particles in 3D, density of states $g(\varepsilon)\propto\sqrt\varepsilon$ . The mean occupation of a state at energy $\varepsilon$ is $e^{-(\varepsilon-\mu)/kT}$ (Maxwell-Boltzmann), $1/(e^{(\varepsilon-\mu)/kT}+1)$ (Fermi-Dirac, one particle per state), or $1/(e^{(\varepsilon-\mu)/kT}-1)$ (Bose-Einstein). The chemical potential $\mu(T)$ is fixed by $N=\int g\,n\,d\varepsilon$ . As $T\to0$ the Fermi gas fills sharply to $E_F$ ; the Bose gas drives $\mu\to0$ at $T_c$ and below it a macroscopic fraction $1-(T/T_c)^{3/2}$ collapses into the ground state. Cool through $T_c$ and watch the condensate spike grow.

Figure 1. Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein occupation versus energy (left), with Bose-Einstein condensation, beside the occupation cells (right): one column per statistic, each row an energy level eps_k (the ticks on the curve's energy axis), each dot a particle, the dot count proportional to g(eps_k) n(eps_k) on a scale shared across the columns. Method: chemical potential solved from fixed N by bisection over Simpson-integrated occupation; condensate fraction in closed form.

temperature tau0.90

statistics

occupied g n

WHAT TO TRY

Switch between Maxwell-Boltzmann, Fermi-Dirac and Bose-Einstein: the same density of states fills completely differently, fermions stack one per state up to the Fermi energy, bosons pile into the ground state.
Lower the temperature tau for the Bose gas: a macroscopic fraction N0/N condenses into the ground state below Tc, the Bose-Einstein condensate spike the readout tracks.
Raise tau for all three: the quantum distributions converge to the classical Maxwell-Boltzmann curve, since at high temperature occupation numbers are small and the statistics no longer matter.