The entropy density s(T) is the derivative of the Helmholtz free energy with respect to temperature,

The Helmholtz free energy density can be expressed in terms of an integral over the density of states.

Here is the energy of phonons with frequency ω and D(ω) is the density of states. This result is discussed in Statistical Physics, Part 1 by Landau and Lifshitz and in the notes on the quantization of the electromagnetic field.

Differentiating with respect to T results in the following expression for the entropy,

The form below uses this formula to calculate the temperature dependence of the entropy 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 in units of s rad^{-1}m^{-3}.

After the 'DoS → s(T)' button is pressed, the density of states is plotted on the left and s(T) is plotted from temperature T_{min} to temperature T_{max} on the right. The data for the s(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 entropy density in units of J K^{-1} m^{-3}.

D(ω) [10^{15} s rad^{-1}m^{-3}]

s(T) [10^{6} J K^{-1} m^{-3}]

ω [10^{12} rad/s]

T [K]

Input: ω [rad/s] D(ω) [s rad^{-1}m^{-3}]

Output: T [K] s(T) [J K^{-1} m^{-3}]

Load a phonon density of states:
Nearest-neighbor mass-spring models: