Why does the Bose curve shoot up near μ?

The ideal-gas Bose function diverges as E approaches μ from above: many bosons attempt to pile into low-lying states. Real gases have interactions and finite volume that regularize this behavior; the simulator omits interactions and simply omits or clips invalid points for clarity.

When should Maxwell–Boltzmann agree with Fermi–Dirac?

When occupancies are small (f ≪ 1), Pauli blocking is negligible and f_FD ≈ e^{β(μ−E)} = f_MB. The ratio f_MB / f_FD displayed in the sidebar approaches 1 in that regime, typically at high energy tails or when the gas is dilute.

Bose–Einstein vs Fermi–Dirac

For noninteracting identical particles in equilibrium with a reservoir at temperature T and chemical potential μ, the mean occupation of a single-particle state of energy E follows quantum statistics. Fermi–Dirac statistics enforce the Pauli exclusion principle: f_FD(E) = 1/(e^{β(E−μ)} + 1) with β = 1/(k_B T). Bose–Einstein statistics allow macroscopic occupation: f_BE(E) = 1/(e^{β(E−μ)} − 1), defined for E > μ in the ideal-gas grand canonical ensemble (divergence as E → μ⁺ signals Bose–Einstein condensation physics not fully captured here). The classical Maxwell–Boltzmann occupation f_MB(E) = e^{β(μ−E)} is the dilute limit where quantum overlaps are negligible. The graph overlays all three at the same T and μ so you can see how raising temperature or shifting μ moves fermionic steps toward exponential tails.

Who it's for: Undergraduates in thermal physics or quantum mechanics courses studying the road from MB statistics to FD and BE gases.

Key terms

Fermi–Dirac distribution
Bose–Einstein distribution
Maxwell–Boltzmann
Chemical potential
Pauli exclusion
Grand canonical ensemble

Occupation vs single-particle energy E (same μ and k_B T). The graph is the main view — raise T or move μ to see FD soften and MB approach FD in the dilute tail.

Live graphs

How it works

Quantum statistics at the same temperature and chemical potential: Pauli exclusion suppresses fermions below μ; bosons pile up toward μ; Maxwell–Boltzmann is the classical no-exclusion limit.

Frequently asked questions

Why does the Bose curve shoot up near μ?: The ideal-gas Bose function diverges as E approaches μ from above: many bosons attempt to pile into low-lying states. Real gases have interactions and finite volume that regularize this behavior; the simulator omits interactions and simply omits or clips invalid points for clarity.
When should Maxwell–Boltzmann agree with Fermi–Dirac?: When occupancies are small (f ≪ 1), Pauli blocking is negligible and f_FD ≈ e^{β(μ−E)} = f_MB. The ratio f_MB / f_FD displayed in the sidebar approaches 1 in that regime, typically at high energy tails or when the gas is dilute.