Variable transformations¶
It is useful to be able to convert derivative operators between the various sets of composition variables. In the relations below, we omit the “bars” and simply write \(n_n,n_p\) for \(\bar{n}_n,\bar{n}_p\). In other words, all of the nucleon densities below are presumed to include nucleons both inside and outside of nuclei.
Converting between (nn,np) and (nB,ne)¶
Since \(n_p=n_e\) and \(n_n=n_B-n_e\),
\[\begin{split}\left(\frac{\partial }{\partial n_B}\right)_{n_e} &=&
\left(\frac{\partial n_n}{\partial n_B}\right)_{n_e}
\left(\frac{\partial }{\partial n_n}\right)_{n_p} +
\left(\frac{\partial n_p}{\partial n_B}\right)_{n_e}
\left(\frac{\partial }{\partial n_p}\right)_{n_n} =
\left(\frac{\partial }{\partial n_n}\right)_{n_p}
\nonumber \\
\left(\frac{\partial }{\partial n_e}\right)_{n_B} &=&
\left(\frac{\partial n_n}{\partial n_e}\right)_{n_B}
\left(\frac{\partial }{\partial n_n}\right)_{n_p} +
\left(\frac{\partial n_p}{\partial n_e}\right)_{n_B}
\left(\frac{\partial }{\partial n_p}\right)_{n_n} =
\left(\frac{\partial }{\partial n_p}\right)_{n_n} -
\left(\frac{\partial }{\partial n_n}\right)_{n_p}\end{split}\]
For second derivatives
\[\begin{split}\left(\frac{\partial^2 }{\partial n_B^2}\right)_{n_e} &=&
\left(\frac{\partial^2 }{\partial n_n^2}\right)_{n_p}
\nonumber \\
\left(\frac{\partial^2 }{\partial n_e\partial n_B}\right) &=&
\left(\frac{\partial^2 }{\partial n_p \partial n_n}\right) -
\left(\frac{\partial^2 }{\partial n_n^2}\right)_{n_p}
\nonumber \\
\left(\frac{\partial^2 }{\partial n_e^2}\right)_{n_B} &=&
\left(\frac{\partial^2 }{\partial n_p^2}\right)_{n_n} -
2\left(\frac{\partial^2 }{\partial n_p \partial n_n}\right) +
\left(\frac{\partial^2 }{\partial n_n^2}\right)_{n_p}\end{split}\]
Converting between (nn,np) and (nB,Ye)¶
Since \(n_p=n_B Y_e\) and \(n_n=n_B(1-Y_e)\),
\[\begin{split}\left(\frac{\partial }{\partial n_B}\right)_{Y_e} &=&
\left(\frac{\partial n_n}{\partial n_B}\right)_{Y_e}
\left(\frac{\partial }{\partial n_n}\right)_{n_p} +
\left(\frac{\partial n_p}{\partial n_B}\right)_{Y_e}
\left(\frac{\partial }{\partial n_p}\right)_{n_n} =
(1-Y_e) \left(\frac{\partial }{\partial n_n}\right)_{n_p} +
Y_e \left(\frac{\partial }{\partial n_p}\right)_{n_n}
\nonumber \\
\left(\frac{\partial }{\partial Y_e}\right)_{n_B} &=&
\left(\frac{\partial n_n}{\partial Y_e}\right)_{n_B}
\left(\frac{\partial }{\partial n_n}\right)_{n_p} +
\left(\frac{\partial n_p}{\partial Y_e}\right)_{n_B}
\left(\frac{\partial }{\partial n_p}\right)_{n_n} =
n_B \left[\left(\frac{\partial }{\partial n_p}\right)_{n_n} -
\left(\frac{\partial }{\partial n_n}\right)_{n_p} \right]\end{split}\]
The inverse transformation is:
\[\begin{split}\left(\frac{\partial }{\partial n_n}\right)_{n_p} =
\left(\frac{\partial }{\partial n_B}\right)_{Y_e}
- \frac{Y_e}{n_B}
\left(\frac{\partial }{\partial Y_e}\right)_{n_B}
\nonumber \\
\left(\frac{\partial }{\partial n_p}\right)_{n_n} =
\left(\frac{\partial }{\partial n_B}\right)_{Y_e}
+ \frac{(1-Y_e)}{n_B}
\left(\frac{\partial }{\partial Y_e}\right)_{n_B}\end{split}\]
This transformation is used in stability() in eos_nuclei.cpp.
There is a Maxwell relation:
\[\frac{\partial f^2}{\partial n_n \partial n_p} =
\frac{\partial f^2}{\partial n_p \partial n_n}\]
which implies
\[\left(\frac{\partial \mu_n}{\partial n_p}\right) =
\left(\frac{\partial \mu_p}{\partial n_n}\right)
\left(\frac{\partial \mu_e}{\partial n_n}\right)\]
or
\[\left(\frac{\partial \mu_p}{\partial n_B}\right)_{Y_e}
- \frac{Y_e}{n_B}
\left(\frac{\partial \mu_p}{\partial Y_e}\right)_{n_B}
=
\left(\frac{\partial \mu_n}{\partial n_B}\right)_{Y_e}
+ \frac{(1-Y_e)}{n_B}
\left(\frac{\partial \mu_n}{\partial Y_e}\right)_{n_B}\]
thus
\[\left(\frac{\partial \mu_p}{\partial n_B}\right)_{Y_e}
=
\frac{Y_e}{n_B}
\left(\frac{\partial \mu_p}{\partial Y_e}\right)_{n_B}
+ \left(\frac{\partial \mu_n}{\partial n_B}\right)_{Y_e}
+ \frac{(1-Y_e)}{n_B}
\left(\frac{\partial \mu_n}{\partial Y_e}\right)_{n_B}\]
This equality is also used in stability() in eos_nuclei.cpp.
Converting between (nn,np) and (nB,Ye) with muons¶
When muons are included, the expressions change, since \(n_p = n_e + n_{\mu}(n_e)\) and \(n_n = n_B - n_e - n_{\mu}(n_e)\),
\[\begin{split}\left(\frac{\partial }{\partial n_B}\right)_{n_e} &=&
\left(\frac{\partial n_n}{\partial n_B}\right)_{n_e}
\left(\frac{\partial }{\partial n_n}\right)_{n_p} +
\left(\frac{\partial n_p}{\partial n_B}\right)_{n_e}
\left(\frac{\partial }{\partial n_p}\right)_{n_n} =
\left(\frac{\partial }{\partial n_n}\right)_{n_p}
\nonumber \\
\left(\frac{\partial }{\partial n_e}\right)_{n_B} &=&
\left(\frac{\partial n_n}{\partial n_e}\right)_{n_B}
\left(\frac{\partial }{\partial n_n}\right)_{n_p} +
\left(\frac{\partial n_p}{\partial n_e}\right)_{n_B}
\left(\frac{\partial }{\partial n_p}\right)_{n_n} =
(1+\chi) \left[
\left(\frac{\partial }{\partial n_p}\right)_{n_n} -
\left(\frac{\partial }{\partial n_n}\right)_{n_p}\right]\end{split}\]
where
\[\chi = \frac{\partial n_{\mu}}{\partial n_e} =
\frac{\partial n_{\mu}}{\partial {\mu}_{\mu}}
\frac{\partial {\mu}_{\mu}}{\partial {\mu}_e}
\frac{\partial {\mu}_{e}}{\partial n_e} +
\frac{\partial {\mu}_{e}}{\partial n_e} =
\frac{\partial n_{\mu}}{\partial {\mu}_{\mu}}
\left(\frac{\partial n_e}{\partial {\mu}_{e}}\right)^{-1}\]
For second derivatives
\[\begin{split}\left(\frac{\partial^2 }{\partial n_B^2}\right)_{n_e} &=&
\left(\frac{\partial^2 }{\partial n_n^2}\right)_{n_p}
\nonumber \\
\left(\frac{\partial^2 }{\partial n_e\partial n_B}\right) &=&
(1+\chi)\left[\left(\frac{\partial^2 }{\partial n_p \partial n_n}\right) -
\left(\frac{\partial^2 }{\partial n_n^2}\right)_{n_p}\right]
\nonumber \\
\left(\frac{\partial^2 }{\partial n_e^2}\right)_{n_B} &=&
\left(1+\chi\right)^2 \left[
\left(\frac{\partial^2 }{\partial n_p^2}\right)_{n_n} -
2\left(\frac{\partial^2 }{\partial n_p \partial n_n}\right) +
\left(\frac{\partial^2 }{\partial n_n^2}\right)_{n_p}\right]\end{split}\]