EoS Inverter

The EoS Inverter module performs the inverse mapping of the Equation of State (EoS). It inverts the mapping between thermodynamic variables \((T, \mu_B, \mu_S, \mu_Q)\) and chosen variables, such as the conserved densities \((s, n_B, n_S, n_Q)\) by solving a coupled nonlinear system, and stores the results in a lookup table (HDF5 or CSV). The inversion target coordinates are fully configurable: any combination of entropy density, energy density, baryon/strangeness/charge densities, or charge fractions can be used as output grid axes. The inversion can be done for EoS tables gridded in up to 4 dimensions. It also works for tables with multiple phase branches, provided the input table contains phase information (this feature is only available up to 3d). The input table itself doesn’t need to carry all four thermodynamic dimensions either: the strangeness and electric-charge dimensions can each be independently omitted via input.has_S/input.has_Q when the tabulated physics doesn’t track them (e.g. a 2D table in \((T,\mu_B)\) for strangeness-neutral, isospin-symmetric matter). See Equation of State for the exact column layout this implies.

This software is available within the MUSES Calculation Engine. See the Usage with the MUSES Calculation Engine for a minimal example of how to use the module.

To use it locally, see the Installation instructions and the Quickstart guide for a simple example.

In QCD matter with conserved baryon number, strangeness, and electric charge, the thermodynamic state is characterized by the pressure

\[P(T,\ \mu_B,\ \mu_S,\ \mu_Q).\]

The pressure encodes the equilibrium thermodynamics, and the conserved densities follow from its derivatives:

\[s(T,\mu) = \left( \frac{\partial P}{\partial T} \right)_{\mu_B,\mu_S,\mu_Q}, \qquad n_i(T,\mu) = \left( \frac{\partial P}{\partial \mu_i} \right)_{T,\mu_{j\neq i}}, \quad i\in\{B,S,Q\}.\]

The susceptibilities (second order derivatives of the pressure) quantify how the system responds to changes in temperature and chemical potentials:

\[\chi_{ij}(T,\mu) = \frac{\partial^2 P}{\partial \mu_i\,\partial \mu_j}, \qquad \chi_{Ti}(T,\mu) = \frac{\partial^2 P}{\partial T\,\partial \mu_i}, \qquad \chi_{TT}(T,\mu) = \frac{\partial^2 P}{\partial T^2}, \quad i,j\in\{B,S,Q\}.\]

These quantities are optional columns in the input table: when present, they are simply passed through to the output table so that downstream modules have access to a consistent set of derivatives. The speed of sound \(c_s^2\) and the stability of a particular point in the EoS are also optional inputs that are passed through to the output table. See the Equation of State for a complete list of supported input columns and their expected units.

Domain coverage and extensions. EoS tables have finite domains in \((T,\mu)\) space. To ensure robustness across the full thermodynamic range requested, the inverter allows augmenting the primary table with controlled backup EoSs (tanh-conformal, conformal, conformal-diagonal, and cd-modified).

  1. Tanh-Conformal EoS: smoothly connects an EOS at finite \(T\) and \(\mu_B\) to a conformal regime via a hyperbolic tangent function.

  2. Conformal EoS: provides a simple scale-free fallback with \(p = e / 3\).

  3. Conformal-Diagonal EoS: decouples densities and guarantees a solution for all inputs.

  4. CD-Modified EoS: a modified conformal-diagonal model for extended coverage.

The user can choose which backup EoSs to enable, and the inverter will automatically fall back to them when the primary table fails to converge. Each backup EoS is automatically matched to the primary table at its boundary: the conformal coefficient \(c\) is set so that \(p_\text{backup}(T_\text{max}) = p_\text{table}(T_\text{max})\), and the chemical-potential scales \(\mu_{B,0},\,\mu_{S,0},\,\mu_{Q,0}\) are chosen to match the pressure increments \(p(T_\text{max},\mu_{i,\text{max}}) - p(T_\text{max},0)\).

These provide smooth extrapolations and guarantee convergence while enforcing thermodynamic stability, causality.

Note

Although the inverter can process EoS tables containing multiple phase branches (stable, metastable, unstable) and will keep them numerically separate throughout the inversion, it is primarily designed and validated for crossover transitions. Within each individual phase branch, the underlying interpolation (multilinear or Delaunay) is itself continuous, so a genuine first-order discontinuity is only as sharp in the inverted output as it is resolved by the input table’s grid spacing.

If the original EoS table samples a first-order transition too coarsely, or the discontinuity spans only a few grid cells, the interpolation step may partially wash out or round off the jump. Users working with EoS tables featuring first-order phase transitions should ensure sufficiently fine grid resolution around the transition region in the input table, and should independently validate that critical features (e.g. the latent heat / density jump) are preserved in the inverted output.

Note

Portions of this documentation were drafted with the assistance of Claude (Anthropic) and have been reviewed by a human.

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