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. This type of PT occurs over a certain range of densities, where phase I occupies a volume fraction \(f\) and phase II 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:

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

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

Hyperbolic tangent interpolation

The interpolation is a well-known method for joining different EoS’s by interpolating them. In this module, the hyperbolic tangent method is used, where the interpolating functions are defined as:

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

where \(x\) is some thermodynamic variable, \(\bar{x}\) is the midpoint of the merging, and \(\Gamma\) is its width. 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’ 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 just like \(Y(x)\), instead of using the thermodynamic relations.

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.