Speed of sound in a multicomponent system
Using \(\varepsilon\) for energy density \(S\) for entropy,
\(s\) for entropy density, and \(\tilde{s}\) for entropy per
baryon, the speed of sound is
\[c_s^2 = \left( \frac{\partial P}{\partial \varepsilon}
\right)_{\tilde{s},\{ N_i \}}
\, .\]
The energy density in the denominator must include the rest mass
contribution to the energy density. In infinite matter, it is useful
to rewrite this derivative in terms of fixed volume rather than fixed
number.
\[c_s^2 = \left( \frac{\partial P}{\partial \varepsilon}
\right)_{S,\{ N_i \}} =
\left( \frac{\partial P}{\partial V} \right)_{S,\{ N_i \}}
\left( \frac{\partial \varepsilon}{\partial V} \right)_{S,\{ N_i \}}^{-1}\]
The second derivative is
\[\left( \frac{\partial \varepsilon}{\partial V} \right)_{S,\{ N_i \}} =
\left[ \frac{\partial (E/V)}{\partial V} \right]_{S,\{ N_i \}} =
-\frac{1}{V} P - \frac{E}{V^2} = - \frac{P+\varepsilon}{V}
= - \frac{T s + \sum_i \mu_i n_i}{V}\]
and first derivative is
\[\begin{split}\left( \frac{\partial P}{\partial V} \right)_{S,\{ N_j \}} &=& -
\left( \frac{\partial \varepsilon}{\partial V} \right)_{S,\{ N_j\}} +
S \left[ \frac{\partial (T/V)}{\partial V} \right]_{S,\{ N_j \}} +
\sum_i
N_i \left[ \frac{\partial (\mu_i/V)}{\partial V} \right]_{S,\{ N_j \}}
\nonumber \\ &=& -
\left( \frac{\partial \varepsilon}{\partial V} \right)_{S,\{ N_j \}} +
S \left[ -\frac{T}{V^2} + \left( \frac{\partial T}{\partial V}
\right)_{S,\{ N_j \}}\right] +
\sum_i
N_i \left[ -\frac{\mu_i}{V^2} +
\left( \frac{\partial \mu_i}{\partial V} \right)_{S,\{ N_j \}}\right]
\nonumber \\ &=& \frac{P + \varepsilon}{V} +
S \left[ -\frac{T}{V^2} - \left( \frac{\partial P}{\partial S}
\right)_{\{N_j\},V}\right] +
\sum_i N_i \left[ -\frac{\mu_i}{V^2} -
\left( \frac{\partial P}{\partial N_i}
\right)_{S,\{N_{j\neq i}\},V}\right] \nonumber \\
&=& - S \left( \frac{\partial P}{\partial S}\right)_{\{n_j\},V}
- \sum_i N_i \left( \frac{\partial P}{\partial N_i}
\right)_{S,\{n_{j\neq i}\},V}\end{split}\]
Putting these two results together gives
\[c_s^2 = \left[s \left( \frac{\partial P}{\partial s}
\right)_{\{n_j\},V} +
\sum_i n_i \left( \frac{\partial P}
{\partial n_i} \right)_{S,\{n_{j\neq i}\},V}\right] \left(
T s + \sum_i \mu_i n_i \right)^{-1}\]
To re-express this in terms of derivatives of the free energy
(which again must include the rest mass contribution),
\[c_s^2 = \left\{s \left[ \frac{\partial (\sum_i \mu_i n_i - f)}
{\partial s} \right]_{\{n_j\},V} +
\sum_i
n_i\left[ \frac{\partial ( \sum_k \mu_k n_k - f)}{\partial n_i}
\right]_{s,\{n_{j\neq i}\},V}\right\} \left(
T s + \sum_i \mu_i n_i \right)^{-1}\]
For the sum over \(k\),
all densities are constant except for \(n_i\), thus
\[\begin{split}\sum_i
n_i \frac{\partial}{\partial n_i}
\left( \sum_k \mu_k n_k - f \right)_{s,\{n_{j\neq i}\},V}
&=& \sum_i n_i \frac{\partial}{\partial n_i}
\left( \sum_{k\neq i} \mu_k n_k + \mu_i n_i -f
\right)_{s,\{n_{j\neq i}\},V} \nonumber \\
&=&
\sum_i \left[ \sum_k n_k \left(\frac{\partial \mu_k }
{\partial n_i}\right)_{s,\{n_{j\neq i}\},V} + \mu_i -
\left(\frac{\partial f}{\partial n_i}\right)_{s,\{n_{j\neq i}\},V}
\right]\end{split}\]
To compute this we need
\[\begin{split}\left(\frac{\partial f}{\partial n_i}\right)_{s,\{n_{j\neq i}\},V} &=&
\left(\frac{\partial f}{\partial n_i}\right)_{\{n_{j\neq i}\},T,V} +
\left(\frac{\partial f}{\partial T}\right)_{n_B,\{n_{j\neq i}\},V}
\left(\frac{\partial T}{\partial n_i}\right)_{\{n_{j\neq i}\},s,V}
= \mu_i - s \left(\frac{\partial T}{\partial n_i}
\right)_{\{n_{j\neq i}\},s,V}
\nonumber \\
\left(\frac{\partial \mu_k}{\partial n_i}\right)_{s,\{n_{j\neq i}\},V} &=&
\left(\frac{\partial \mu_k}{\partial n_i}\right)_{\{n_{j\neq i}\},T,V} +
\left(\frac{\partial \mu_k}{\partial T}\right)_{n_i,\{n_{j\neq i}\},V}
\left(\frac{\partial T}{\partial n_i}\right)_{\{n_{j\neq i}\},s,V}
= f_{n_i n_k} + f_{n_k T}
\left(\frac{\partial T}{\partial n_i}\right)_{\{n_{j\neq i}\},s,V}\end{split}\]
which requires
\[\left(\frac{\partial T}{\partial n_i}\right)_{\{n_{j\neq i}\},s,V}
= -\left(\frac{\partial s}{\partial n_i}\right)_{\{n_{j\neq i}\},T,V}
\left(\frac{\partial s}{\partial T}\right)_{\{n\},V}^{-1}
= -f_{n_i T}/f_{TT}\]
Finally, we get
\[c_s^2 = \left\{
- \left(\frac{s}{f_{TT}}\right) \left( \sum_i n_i f_{n_i T}+s \right)
+ \sum_i n_i \left[ \sum_k n_k \left(f_{n_i n_k}- f_{n_k T}
f_{n_i T} f_{TT}^{-1}\right)
- s f_{n_i T} f_{TT}^{-1}\right]
\right\} \left(
T s + \sum_i \mu_i n_i \right)^{-1}\]
and
\[c_s^2 = \left[
\sum_i \sum_k n_i n_k \left(f_{n_i n_k}- f_{n_k T}
f_{n_i T} f_{TT}^{-1}\right)
- 2\sum_i s n_i f_{n_i T} f_{TT}^{-1}
- s^2 f_{TT}^{-1} \right] \left(
T s + \sum_i \mu_i n_i \right)^{-1}\]
Note that, when applying this expression, one must be consistent about
the free energy which one differentiates and the densities and
chemical potentials which are used. See Chemical potentials for
more information regarding this issue.
In terms of baryon density and electron fraction
In the expression above there are three terms in square brackets, the
double sum, the single sum, and the final term. In nondegenerate
matter, the double and single sums involve significant cancelations.
To resolve these, it is helpful to re-express these sums in terms of
\(n_B\) and \(Y_e\). Since our degrees of freedom are
neutrons and protons, one finds
\[\begin{split}\begin{eqnarray}
&{\cal{E}}_1 \equiv \sum_i \sum_k n_i n_k \left(f_{n_i n_k}- f_{n_k T}
f_{n_i T} f_{TT}^{-1}\right) = & \nonumber \\
& n_n^2 \left(f_{n_n n_n}- f_{n_n T}^2
f_{TT}^{-1}\right) + 2 n_n n_p \left(f_{n_n n_p}- f_{n_n T}
f_{n_p T} f_{TT}^{-1}\right) +
n_p^2 \left(f_{n_p n_p}- f_{n_p T}^2
f_{TT}^{-1}\right) = & \nonumber \\
& n_B^2 \left(\frac{\partial \mu_n}{\partial n_B}\right)_{Y_e,T} +
n_B^2 \left(\frac{\partial s}{\partial n_B}\right)_{Y_e,T}^2
\left(\frac{\partial s}{\partial T}\right)_{n_B,Y_e}^{-1} + &
\nonumber \\
& n_B Y_e (1-Y_e)
\left(\frac{\partial \mu_n}{\partial Y_e}\right)_{n_B,T} +
n_B Y_e^2 \left(\frac{\partial \mu_p}{\partial Y_e}\right)_{n_B,T} &
\end{eqnarray}\end{split}\]
and
\[{\cal{E}}_2 \equiv \sum_i n_i f_{n_i T} f_{TT}^{-1} =
n_B \left(\frac{\partial s}{\partial n_B}\right)_{Y_e,T}
\left(\frac{\partial s}{\partial T}\right)_{n_B,Y_e}^{-1}\]
Taking advantage of these two expressions
\[c_s^2 = \left[ {\cal{E}}_1 - 2 s {\cal{E}}_2
- s^2 f_{TT}^{-1} \right] \left(
T s + \sum_i \mu_i n_i \right)^{-1}\]