Physics Background
------------------

The pressure is given by

.. math::

    P = -\Omega^{\mathrm{tot}} \equiv - \sum_{f=u,d,s} \left( \Omega_f^{(0)}(m_f,\mu_f) + \Omega_f^{(1)}(m_f,\mu_f) \right),

where the sum runs over the quark flavors :math:`f`.

The total thermodynamic potential is taken from arXiv:1507.08941 [hep-ph]:

.. math::

    \Omega_f^{(0)}  -\frac{N_c}{12\pi^2}\left[\mu_f k_f\left(\mu_f^2 - \frac{5}{2}m_f^2\right) + \frac{3}{2}m_f^4\ln\left(\frac{\mu_f+k_f}{m_f}\right)\right]

.. math::

    \begin{aligned}
    \Omega_f^{(1)} &=
    \frac{\alpha_s N_G}{16\pi^3}\Bigg\{3\Biggr[m_f^2 \ln\left(\frac{\mu_f+k_f}{m_f}\right) - \mu_f k_f\bigg]^2- 2k_f^4  \\
    & +m_f^2\left[    6\ln\left(\frac{\Lambda}{m_f}\right)+4\right]\left[    \mu_f k_f- m_f^2 \ln\left(\frac{\mu_f+k_f}{m_f}\right)\right]\Biggr\}
    \end{aligned}

Here,

- :math:`N_c` is the color degeneracy,
- :math:`N_G = N_c^2 - 1` is the number of gluons,
- :math:`\Lambda` is the renormalization scale,
- :math:`\Omega^{(0)}` is the free-gas contribution,
- :math:`\Omega^{(1)}` is the next-to-leading-order contribution,
- :math:`k_f \equiv \sqrt{\mu_f^2 - m_f^2}` for each flavor :math:`f`.

The running coupling is

.. math::

    \alpha_s(\Lambda)=\frac{4\pi}{\beta_0 \bar{L}}\left[1- 2\frac{\beta_1}{\beta_0^2}\frac{\ln \bar{L}}{\bar{L}}\right]

The strange-quark mass as a function of the strong coupling is

.. math::

    m_s(\Lambda) = \hat{m}_s \left(\frac{\alpha_s}{\pi}\right)^{4/9} \left[ 1 + 0.895062\frac{\alpha_s}{\pi} \right]

where

.. math::

    \bar{L} = 2\ln\left(\frac{\Lambda}{\Lambda_{\overline{\mathrm{MS}}}}\right),
    \qquad
    \beta_0 = 11 - \frac{2}{3}N_f,
    \qquad
    \beta_1 = 51 - \frac{19}{3}N_f.

The scale :math:`\Lambda_{\overline{\mathrm{MS}}}` and the invariant mass :math:`\hat{m}_s` are fixed according to the same definitions used in arXiv:1507.08941 [hep-ph].

The renormalization scale is chosen as

.. math::

    \Lambda = \sqrt{ (2 \pi T)^2 + \left( C_B \mu_B+ C_Q\mu_Q + C_S \mu_S \right)^2 }.

where the :math: `C's` are constants that can be set by the user. The default values are :math:`C_B = 1/3`, :math:`C_Q = 1/6`, and :math:`C_S = 0`.
The energy density is given by the Gibbs-Duhem relation at vanishing temperature:

.. math::

    \epsilon = n_B\mu_B + n_S\mu_S + n_Q\mu_Q - P.

The isospin asymmetry coefficient is defined as

.. math::

    \delta_I = 1 - 2Y_Q + Y_S.

In terms of the quark densities, the charge fraction can be written as

.. math::

    Y_Q =  \frac{2n_u - n_d - n_s}{n_u + n_d + n_s}.