PX284 - N2 - boson gases

photon gas

\begin{matrix} E = ℏ ω \\ ω = c k \\ g (ω) d ω = 2 \frac{V ω^{2}}{2 π^{2} c^{3}} d ω \\ U = \int_{0}^{\infty} ℏ ω g (ω) f_{B E} (ω) d ω = \int_{0}^{\infty} \frac{ℏ V}{π^{2} c^{3}} \frac{ω^{3}}{\exp (β ℏ ω) - 1} d ω \end{matrix}

\begin{matrix} g (ω) d ω = 3 \times \frac{V ω^{2}}{2 π^{2} v_{s}^{3}} d ω \\ U = \int_{0}^{\infty} \frac{9 N ℏ}{ω_{D}^{3}} \frac{ω^{3}}{\exp (β ℏ ω) - 1} d ω \\ ω_{D} = {(\frac{6 N π^{2} v_{s}^{3}}{V})}^{1 / 3} \end{matrix}

N = \int_{0}^{\infty} (2 s + 1) \frac{V}{(2 π)^{2}} {(\frac{2 m}{ℏ})}^{3 / 2} \frac{E^{1 / 2}}{\exp (β (E - μ)) - 1} d E

where, $μ < 0$