Parameters

The 3D Ising 2D T’-Expansion Scheme (TExS) module includes a range of input parameters that control the construction of the equation of state (EoS). These parameters include grid definitions, numerical discretization choices, and mapping parameters used to embed 3D Ising critical behavior into QCD thermodynamics. The output produced by the module depends directly on the selected parameter configuration.

This section provides a detailed description of the parameters used by the module.

As described in the Quick Start Guide, running the Ising-2DTExS EoS module requires the user to provide a config.yaml file. This file contains all of the configuration information required to run the module, including numerical grid settings, mapping parameters, and output options.

The structure and validation rules of the configuration file follow the OpenAPI 3.0.0 specification provided with the module.

Each time the module is executed, it reads the user-provided config.yaml file and verifies that the inputs conform to the OpenAPI specification. If the configuration does not satisfy the specification, execution will fail. When this occurs, the user should inspect the program logs to identify the configuration errors.

It is not necessary to specify every parameter explicitly in the configuration file. If a parameter is omitted, the module automatically assigns its default value. The default values for each parameter are defined in the OpenAPI specification.

Upon successful execution, the module generates CSV output files in the output directory. The exported columns depend on the selected switchers: only the quantities requested in config.yaml are written to the thermodynamic table, while the transition line is exported separately.

The following sections provide an overview of the module input parameters, internal parameters, and output files.

Input Parameters

Input parameters define the thermodynamic grid, numerical discretization level, and the mapping between the 3D Ising model variables and the QCD thermodynamic variables.

Simulation Mode

The module can operate in two different modes.

EoS Parameters

All the following parameters are specified in the parameters section of the configuration file.

Grid type

The type of grid used to compute the thermodynamic quantities is specified by the grid_type parameter of the parameters section in the configuration file. The module supports several grid types, including:

RegularUniformGrid: A uniform grid in both temperature and baryon chemical potential.

RegularGridwTCmuBC: A uniform grid set to be centered exactly on the critical point location.

UniformT_NonUniformmuB: A uniform grid in temperature and a non-uniform grid in baryon chemical potential.

UniformmuB_NonUniformT: A uniform grid in baryon chemical potential and a non-uniform grid in temperature.

RegularGridZoom: A uniform grid with increased resolution around the expected location of the critical point, with a width parameter controlling the zoom region

Grid limits

The temperature and baryon chemical potential ranges over which the EoS is computed are then defined, still as part of the parameters section of the configuration file, by specifying the minimum and maximum values for each variable, as well as the base step size for the grid.

Category	Parameter	Default	Description
`T`	`min`	`True`	Minimum temperature in MeV
	`max`	`True`	Maximum temperature in MeV
	`step`	`True`	Base step size in temperature in MeV
`muB`	`min`	`True`	Minimum temperature in MeV
	`max`	`True`	Maximum temperature in MeV
	`step`	`True`	Base step size in temperature in MeV

3D Ising to QCD Mapping Parameters

These parameters define how the 3D Ising universality class variables \((r,h)\) are mapped onto the QCD thermodynamic variables \((T,\mu_B)\), to be specified when using mode: PAR in the parameters section of the configuration file.

More details about the mapping are given in the Mapping Ising to QCD section of the Physics documentation.

Category	Parameter	Description
`mapping`	`muBC`	Location of the critical point in baryon chemical potential, in MeV
	`T_crit`	Critical temperature, in MeV
	`alpha12`	Rotation angle between the Ising axes and the QCD axes, in degrees
	`w`	Controls the strength and extent of the critical region
	`rho`	Controls the anisotropy of the critical scaling region

These parameters determine the orientation and scaling of the critical region in the QCD phase diagram.

Critical Region Zoom

When using RegularGridZoom, the module increases the resolution around the expected location of the critical point.

Category	Parameter	Description
`critical_region`	`width`	Half-width of the zoom region centered at the critical point
	`T_width`	Fine temperature step used inside the zoom region
	`muB_width`	Fine baryon chemical potential step used inside the zoom region

The zoom region is defined as

\[T \in [T_{\mathrm{crit}} - \mathrm{width},\; T_{\mathrm{crit}} + \mathrm{width}]\]

\[\mu_B \in [\mu_{BC} - \mathrm{width},\; \mu_{BC} + \mathrm{width}]\]

Thermodynamics Switchers

The following switchers allow users to select which thermodynamic quantities are computed and stored in the output files. Each switcher can be set to True or False in the configuration file.

Category	Input Parameter	Default	Description
`switchers`	`get_transition_line`	`False`	Store selected thermodynamic quantities along the transition line in a separate output file
	`get_B_dens`	`True`	Store baryon density (`B_dens`) in the output
	`get_E_dens`	`True`	Store energy density (`E_dens`) in the output
	`get_P`	`True`	Store pressure (`P`) in the output
	`get_s_dens`	`True`	Store entropy density (`s_dens`) in the output
	`get_chi2B`	`False`	Calculate and store the second-order baryon number susceptibility (\(\chi_2^B\))
	`get_dnBdT`	`False`	Calculate and store the temperature derivative of the baryon number density
	`get_dsdT`	`False`	Calculate and store the temperature derivative of the entropy density
	`get_cs2`	`False`	Calculate and store the squared isentropic speed of sound
	`get_cv`	`False`	Calculate and store the specific heat at constant volume
	`get_TraceAnomaly`	`False`	Calculate and store the trace anomaly (interaction measure)
	`get_chi3B`	`False`	Calculate and store the third-order baryon number susceptibility (\(\chi_3^B\))
	`get_chi4B`	`False`	Calculate and store the fourth-order baryon number susceptibility (\(\chi_4^B\))

Example Configuration File

The following is an example config.yaml file for running the Ising-2DTExS module.

mode: LAT

parameters:
  grid_type: RegularGridZoom

  T:
    min: 30.0
    max: 800.0
    step: 1.0

  muB:
    min: 0.0
    max: 700.0
    step: 1.0

  mapping:
    muBC: 420.0
    T_crit: 133.356
    alpha12: 90.0
    w: 4.25
    rho: 1.0

  critical_region:
    width: 10.0
    muB_width: 0.1
    T_width: 0.1

switchers:
 get_transition_line: false
 get_B_dens: true
 get_E_dens: true
 get_P: true
 get_s_dens: true
 get_chi2B: true
 get_dnBdT: true
 get_dsdT: true
 get_cs2: true
 get_cv: true
 get_TraceAnomaly: true
 get_chi3B: true
 get_chi4B: true

Output Parameters

After a successful execution, the module generates tabulated thermodynamic quantities describing the QCD EoS. The output is written in CSV format and stored in the output directory.

The generated files depend on which quantities are enabled in the switchers section of the configuration file. Disabling unnecessary quantities can reduce runtime and file size.

Equation of State Output

File: Ising_TExS_thermo.csv

This file contains thermodynamic quantities evaluated on the specified \((T,\mu_B)\) grid. Each row corresponds to a single grid point in temperature and baryon chemical potential. Which columns are present depends on the switchers selected in the configuration file: T and muB are always exported, while all other columns are only written if the corresponding switcher is enabled.

Column	Unit	Description
`T`	MeV	Temperature
`muB`	MeV	Baryon chemical potential
`B_dens`	MeV³	Baryon number density
`E_dens`	MeV⁴	Energy density
`P`	MeV⁴	Pressure
`s_dens`	MeV³	Entropy density
`chi2B`	MeV²	Second-order baryon number susceptibility \(\chi_2^B\)
`dB_densdT`	MeV²	Temperature derivative of the baryon number density
`ds_densdT`	MeV²	Temperature derivative of the entropy density
`cs2`	–	Squared isentropic speed of sound
`cv`	MeV³	Specific heat at constant volume
`TraceAnomaly`	MeV⁴	Trace anomaly (interaction measure)
`chi3B`	MeV	Third-order baryon number susceptibility \(\chi_3^B\)
`chi4B`	–	Fourth-order baryon number susceptibility \(\chi_4^B\)

The thermodynamic quantities are computed using the 3D Ising-based T’-Expansion Scheme, which embeds critical behavior into the QCD equation of state.

Example Output

Below is a sample of the output produced by the module.

T,mu_B,P,s_dens,E_dens,B_dens,c_s,chi2B
30,0,23361,5364.58,137576,0,0.185573,2.08927e-09
30,1,23361,5364.58,137576,2.08965e-09,0.185573,2.09041e-09
30,2,23361,5364.58,137576,4.18157e-09,0.185573,2.09382e-09
30,3,23361,5364.58,137576,6.27805e-09,0.185573,2.09952e-09
30,4,23361,5364.58,137576,8.38137e-09,0.185573,2.1075e-09
30,5,23361,5364.58,137576,1.04938e-08,0.185573,2.1178e-09
30,6,23361,5364.58,137576,1.26177e-08,0.185573,2.13041e-09

Each row corresponds to a unique pair of \((T,\mu_B)\) values from the computational grid.

Notes on Units

All thermodynamic quantities are reported in natural units consistent with the conventions used internally by the module.

In particular:

Temperature \(T\) and baryon chemical potential \(\mu_B\) are given in MeV.
Thermodynamic quantities such as pressure, energy density, and entropy density are reported in units listed in the Units section.

Users should ensure consistent unit conversion when interfacing the output with hydrodynamic simulations or astrophysical modeling frameworks.

Internal Parameters

The module uses parameterizations of lattice QCD observables at zero chemical potential. These parameterizations provide the baseline thermodynamics onto which the Ising critical behavior is embedded.

Lattice Pressure

The parameterized lattice pressure at zero chemical potential is

\[P_{0,\mathrm{lat}}(T) = \frac{a_0 + a_1 x + a_2 x^2 + a_3 x^3 + a_4 x^4 + a_5 x^5} {b_0 + b_1 x + b_2 x^2 + b_3 x^3 + b_4 x^4 + b_5 x^5}\]

where

\[x = \frac{154}{T}\]

The coefficients \(a_i\) and \(b_i\) are determined from fits to lattice QCD results.

\(a_0\)	\(a_1\)	\(a_2\)	\(a_3\)	\(a_4\)	\(a_5\)	\(b_0\)	\(b_1\)	\(b_2\)	\(b_3\)	\(b_4\)	\(b_5\)
7.53891	-6.18858	-5.37961	7.08750	0.977970	0.0302636	2.24530	6.02568	15.3737	19.6331	10.2400	0.799479

Lattice Baryon Number Susceptibility

The lattice baryon number susceptibility at zero chemical potential is parameterized as

\[\chi_{2,\mathrm{lat}}^B(T) = \left(\frac{2m_p}{\pi x}\right)^{3/2} \frac{e^{-m_p/x}}{1+\left(\frac{x}{d_1}\right)^{d_2}} + d_3 \frac{e^{-d_4^2/x^2 - d_5^4/x^4}} {1+\left(\frac{x}{d_1}\right)^{-d_2}}\]

where

\[x = \frac{T}{200\,\mathrm{MeV}}\]

and \(m_p\) is the proton mass.

The coefficients of the parameterized \(\chi_{2,\mathrm{lat}}^B(T)\) for \(25\,\mathrm{MeV} \leq T \leq 800\,\mathrm{MeV}\) are

\(d_1\)	\(d_2\)	\(d_3\)	\(d_4\)	\(d_5\)
0.73	11.19	0.32	0.20	0.69

Lattice \(\kappa_2\) Coefficients

The parameterized \(T'\) lattice coefficient \(\kappa_2^{BB}\) is given by

\[\kappa_{2}^{BB}(T) = \frac{A_1 b_0 x + a_2 x^2 + a_3 x^3 + A_2 x^4} {b_0 + b_1 x + x^2}\]

where

\[x = \frac{T}{200\,\mathrm{MeV}}\]

with \(A_1 = 0.035\) and \(A_2 = 1.47\).

The best-fit parameters for \(\kappa_2^{BB}(T)\) are listed below.

\(a_2\)	\(a_3\)	\(b_0\)	\(b_1\)
0.652	-2.60	21.4	-9.81

Transition-Line Output

File: transition_line.csv

When get_transition_line is enabled in the configuration file, the transition line is written to a separate CSV file rather than being folded into the main thermodynamic table. The file stores the tabulated transition curve, in terms of the baryon chemical potential \(\mu_B\) and the corresponding transition temperature \(T_c(\mu_B)\).

In addition to muB and Tc, the file may also contain any of the thermodynamic quantities listed in the Equation of State Output section above, evaluated along the transition line rather than on the full \((T,\mu_B)\) grid. These additional columns follow the exact same column ordering, naming, and unit conventions as the main thermodynamic table, and are controlled by the same switchers used to select the quantities exported in Ising_TExS_thermo.csv.