Quantum Gas Statistics

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

Vary each control and watch the rail readouts respond.

Compare the diagnostic plot against the live scene.