Bose–Einstein condensation (ideal Bose gas)

Example: ideal Bose gas in $d=3$

Consider $N$ noninteracting bosons of mass $m$ in a cubic box of volume $V$ (periodic boundary conditions). Thermodynamic behavior is described via the canonical ensemble (or, equivalently for this system, a grand-canonical viewpoint).

Key scales and formulas

Define the thermal de Broglie wavelength

For the ideal Bose gas, the excited-state density saturates as the fugacity approaches $1$:

Hence the critical temperature $T_c$ is determined by $n=n_{\mathrm{ex}}(T_c)$:

For $T<T_c$, a macroscopic fraction occupies the one-particle ground state (the condensate):

Thermodynamic consequences

• The pressure (log-partition density) for the ideal Bose gas becomes independent of density below $T_c$ (extra particles go into the condensate rather than increasing pressure).
• The internal energy is carried by excited modes, giving $U\propto V T^{5/2}$ for $T<T_c$, so the $C_V$ scales like $V T^{3/2}$ in the condensed phase.
• The transition can be diagnosed through standard phase transition indicators (non-analyticity in thermodynamic potentials in the thermodynamic limit).

Prerequisites / cross-links

• canonical partition function , statistical free energy , ensemble averages
• temperature , thermodynamic entropy
• pressure