Bose gas

Example

An ideal Bose gas is a system of non-interacting identical bosons obeying Bose–Einstein statistics. Take spinless bosons of mass $m$ in a box of volume $V$ in three dimensions, at temperature $T$ and chemical potential $\mu < 0$. Write $\beta = 1/(k_B T)$, $z = e^{\beta \mu} \in (0, 1)$ for the fugacity, and $$\lambda = \frac{h}{\sqrt{2 \pi m k_B T}}$$ for the thermal wavelength, where $h$ is the Planck constant. In the thermodynamic limit, replacing the sum over states by an integral over momenta gives the density $$n = \frac{1}{h^3} \int_{\R^3} \frac{d^3 p}{z^{-1} e^{\beta \abs{p}^2 / 2m} - 1} = \frac{g_{3/2}(z)}{\lambda^3}, \qquad g_s(z) = \sum_{k \ge 1} \frac{z^k}{k^s}.$$ Check: expand $1/(z^{-1}e^{x} - 1) = \sum_{k \ge 1} z^k e^{-kx}$ and use the Gaussian integral $\frac{1}{h^3}\int e^{-k\beta \abs{p}^2/2m} d^3 p = k^{-3/2} \lambda^{-3}$. The function $g_s$ is the polylogarithm $\operatorname{Li}_s$.