The particle number density $n(T)$ is minus the derivative of the grand potential density $\phi$ with respect to the chemical potential $\mu$ and can be expressed in terms of an integral over the density of states. For the case where $\mu=0$,
\begin{equation} \large n = -\frac{\partial \phi}{\partial \mu} = \int \limits_{-\infty}^{\infty} D(E) \frac{1}{\exp\left( \frac{E}{k_BT}-1 \right)} dE. \end{equation}The particle number density is the density of states times the Bose-Einsten factor summed over all energies. Written in terms of the density of states in frequency, $E=\hbar\omega$, $D(E)dE=D(\omega)d\omega$,
\begin{equation} \large n = \int \limits_{0}^{\infty} D(\omega) \frac{1}{\exp\left( \frac{\hbar\omega}{k_BT}-1 \right)} d\omega. \end{equation}This result is discussed in Statistical Physics, Part 1 by Landau and Lifshitz and in the notes on thermodynamic properties of noninteracting bosons.
The form below uses this formula to calculate the temperature dependence of the particle number density from tabulated data for the density of states. The density of states data is input as two columns in the textbox at the lower left. The first column is the angular-frequency ω in rad/s. The second column is the density of states. The units of the density of states depends on the dimensionality: s/m for 1d, s/m² for 2d, and s/m³ for 3d.
After the 'DoS → n(T)' button is pressed, the density of states is plotted on the left and n(T) is plotted from temperature Tmin to temperature Tmax on the right. The data for the n(T) plot also appear in tabular form in the lower right textbox. The first column is the temperature in Kelvin and the second column is the particle number density in units of m-1, m-2, or m-3 depending on the dimensionality.
| D(ω) | n(T) [1010] | |||
ω [1015 rad/s] |
T [K] | |||
|
Input: [ω D(ω)] |
Output: [T n(T)] | |||
Photons in vacuum: