Density of states → grand potential density $\phi$(T)

The thermodynamic grand potential $\Phi(T)$ can be determined from the grand canonical partition function, $ \Phi = U - TS - \mu N = -k_B T ~ \ln(Z_{gr}) $. Here $U$ is the internal energy, $S$ is the entropy and $N$ is the average number of bosons in the system. For photons this gives: \begin{equation} \large \Phi = k_B T ~ \sum_i \ln \left[ 1- \exp \left( \frac{-\epsilon_i}{k_B T} \right) \right] \end{equation} With the chemical potential $\mu = 0$ and $\epsilon_i = \hbar \omega_i$.

This sum can be approximated by an integral over the density of states $D(\omega)$. The density of states is defined per unit volume so the grand potential density is then defined as: \begin{equation} \large \phi = \frac{\Phi}{V} = k_B T \int_{-\infty}^{\infty} D(\omega) ~ \ln \left[ 1 - \exp \left( \frac{-\hbar \omega}{k_B T} \right) \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 grand potential 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 → φ(T)' button is pressed, the density of states is plotted on the left and $\phi(T)$ is plotted from temperature $T_{min}$ to temperature $T_{max}$ on the right. The data for the $\phi(T)$ plot also appears in tabular form in the lower right textbox. The first column is the temperature in Kelvin and the second column is the grand potential density in units of J/m, J/m² or J/m³ depending on the dimensionality.

    $\phi$(T) [10-9]

ω [1015 rad/s]

T [K]

Tmin:  K  Tmax:  K

Input: [ω D(ω)]

Output: [Tn(T)]

Photons in vacuum: