Physics

At the current moment, the module supports:

Maxwell phase transition

The Maxwell phase transition requires only two EoS files, specified via the eos_input_file options under the model1_BetaEq_files and model2_BetaEq_files sections. This type of phase transition (PT) occurs when the baryon chemical potentials and pressures cross:

\[\mu_B^I = \mu_B^{II} \qquad P^I = P^{II}\]

While these quantities are continuous, others such as the baryon density (\(n_B\)), energy density (\(\varepsilon\)), charge chemical potential (\(\mu_Q\)), and charge density (\(n_Q\)) are discontinuous. To use this phase transition, set the option synthesis_type to ‘maxwell’.

Gibbs phase transition

The Gibbs phase transition requires two beta-equilibrated EoS files, the pure leptonic EoS, and the two gridded EoS files without leptons (i.e., the EoS files that come directly from the Nuclear physics module). All of these are specified in the eos_input_file options. The gridded EoS files must be a \((\mu_B, \mu_Q)\) grid at a single fixed temperature (strangeness-neutral rows only, \(\mu_S = 0\)) - the module raises an error if more than one temperature value is present. The leptonic EoS file is read as a function of the electron chemical potential \(\mu_e\) at that same fixed temperature. This is a different grid convention from the one used by the statistical mixture method, described below. This type of PT occurs over a certain range of densities, where phase II occupies a volume fraction \(f\) (denoted vol_M2 in the code and in the transition.csv output) and phase I occupies \(1-f\). In this case, pressure, baryon chemical potential, and charge chemical potentials match:

\[\mu_B^I = \mu_B^{II} \qquad \mu_Q^I = \mu_Q^{II} \qquad P^I = P^{II}\]

and the net electric charge must be zero:

\[(1-f) n_Q^I + f\, n_Q^{II} + n_Q^{leptons} = 0\]

To use this phase transition, set the option synthesis_type to ‘gibbs’.

Equation of State interpolation

The interpolation is a well-known method for joining different EoS’s by interpolating between them. In this module, the options to use either a hyperbolic-tangent, a Gaussian or a bump function are implemented,: where the interpolating functions are defined as:

Hyperbolic-tangent function:

\[f_\pm(x) = \frac{1}{2} \left[ 1 \pm \tanh \left( \frac{x - \bar{x}}{\Gamma} \right) \right],\]

Gaussian-like function:

\[\begin{split}f_-(x) = \begin{cases} 1 & \text{if } x \leq \bar{x} \\ \exp\left( -\dfrac{(x - \bar{x})^n}{\Gamma^n} \right) & \text{if } x > \bar{x} \end{cases} \quad, \qquad f_+(x) = 1 - f_-(x)\end{split}\]

Bump function:

\[f_-(x) = 1 - \exp\left( -\dfrac{ \bar{x}}{x}\right)^n \qquad f_+(x) = 1 - f_-(x)\]

Smoothstep function:

\[\begin{split}f_-(x) = \begin{cases} 1 & \text{if } x \leq \bar{x} - \Gamma \\ 1 - \left[ 3t^2 - 2t^3 \right] & \text{if } \bar{x} - \Gamma < x < \bar{x} + \Gamma \\ 0 & \text{if } x \geq \bar{x} + \Gamma \end{cases} \quad, \qquad f_+(x) = 1 - f_-(x), \qquad t = \dfrac{x - (\bar{x} - \Gamma)}{2\Gamma}\end{split}\]

where \(f_+\) and \(f_-\) are the functions used to interpolate between the two EoS’s, \(\bar{x}\) is the midpoint (start) of the merging for the hyperbolic-tangent and smoothstep (Gaussian and Bump) functions, and \(\Gamma\) controls the width. \(x\) is some thermodynamic variable to be chosen.

The function \(Y(x)\) to be interpolated can be chosen and it is defined as:

\[Y(x) = Y^I f_- (x) + Y^{II} f_+ (x)\]

The options for the variables \(Y(x)\) are the following:

\[\varepsilon(n_B), \quad P(n_B), \quad P(\mu_B), c_s^2(n_B)\]

Other thermodynamic quantities are obtained from standard thermodynamic relations, such as

\[P= n_B^2 \frac{\partial \varepsilon/n_B}{\partial n_B} \text{,} \quad n_B = \frac{\partial P}{\partial \mu_B} \text{and} \mu_B= \frac{\varepsilon+ P}{n_B}\]

except in the speed of sound interpolation, where we integrate the EoS upwards, using

\[n_{B, i+1} = n_{B, i} + \Delta n_B\]

\[\varepsilon_{i+1} = \varepsilon_i + \Delta n_B \left( \frac{\varepsilon_i + P_i}{n_{B, i}} \right)\]

\[P_{i+1} = P_i + c_s^2(n_{B, i}) \Delta \varepsilon\]

following arXiv:2106.03890.

To use the interpolation, set synthesis_type to ‘hyperbolic-tangent’, ‘gaussian’, ‘bump’ or ‘smoothstep’ in the config file, and use the variables in the interpolation_method section. The option use_thermodynamic_consistency is enabled by default, and it is highly recommended. If it is disabled by the user, all thermodynamic variables will be obtained by the same equation as for \(Y(x)\), which is equivalent to setting the rearrangement terms to zero.

Additionally, the parameters \(\bar{X}\) and \(\Gamma\) can be computed from the endpoints of the interpolation range, or they can be set by the user. They will be computed automatically if auto_profile_from_bounds is enabled. The user can define a small epsilon to avoid numerical problems, such that the boundaries are defined as:

\[f_-(x_{\text{start}}) = 1 - \epsilon, \qquad f_-(x_{\text{end}}) = \epsilon\]

Attachment method

To simply attach the EoS, set synthesis_type to ‘attach’ and define the attach_variable and attach_value. For example, if the user chooses attach_variable=pressure and attach_value=10, then the output will contain the first EoS at \(P < 10\) MeV fm^-3 and the second EoS at \(P > 10\) MeV fm^-3.

Statistical Mixture

The statistical mixture requires only two EoS files, specified via the eos_input_file options under the model1_Grid_files and model2_Grid_files sections - the leptonic EoS is not used. These files use a different grid convention from the one required by the Gibbs construction above: they must be a \((T, \mu_B)\) grid spanning multiple temperature values (the method solves for the mixture point-by-point over the full temperature-baryon chemical potential mesh), and rows are not filtered by \(\mu_S\). The method defines a grand potential density (negative pressure) \(\omega = -P\) that combines the grand potential density from EoS 1 \(\omega_1 = -P_1\) and EoS 2 \(\omega_2 = -P_2\)

\[\omega = p\,\omega_1 + (1-p)\,\omega_2 + a\,\left[p\,\ln p + (1-p)\,\ln(1-p)\right] + p\,(1-p)\,b,\]

where \(a = T/\Delta V\) and \(b = 2\,T_c/\Delta V\). \(\Delta V\) is a pseudo-volume that the user can tune to control the stiffness of the phase transitions, and \(T_c\) is the temperature of the critical point that can be set by the user. For each \((T, \mu_B)\) point, the mixing weight \(p\) is obtained by minimising \(\omega\) over \(p \in (0, 1)\), keeping only stationary points with \(\partial^2\omega/\partial p^2 > 0\) (local stability); if no stable interior minimum exists, the construction falls back to the lower-\(\omega\) (higher-pressure) pure phase, \(p \in \{0, 1\}\). All the thermodynamics can be calculated following the grand potential density defined above, following arXiv:2601.07987.

If a given \((T, \mu_B)\) point lies inside the domain of only one of the two input tables, that table is copied through unchanged (no mixing), which can leave a discontinuity at the edge of the overlap region.