Van der Waals-like Equation of State

• The van der Waals-like equation of state is a thermodynamic model that extends the ideal gas law by incorporating short‐range repulsion and long‐range attraction to capture liquid–gas coexistence and critical behavior.
• It is mathematically structured through canonical and grand canonical formulations, enabling derivation of critical points, coexistence curves, and characterization of fluctuation phenomena.
• Extensions such as quantum corrections and multicomponent formulations enhance its applicability in nuclear physics, cosmology, and planetary science for more accurate phase transition modeling.

A van der Waals-like equation of state (vdW-EoS) is a class of analytic thermodynamic models that extend the ideal-gas equation by incorporating explicit contributions from both short-range repulsive and long-range attractive interactions between particles. The canonical vdW-EoS captures liquid–gas coexistence, the existence of a critical point, and nontrivial fluctuation behavior. Many generalizations exist, ranging from quantum statistical extensions to multi-parameter interpolating forms relevant to nuclear, soft-matter, and cosmological contexts.

1. Mathematical Structure and Thermodynamic Interpretation

The original van der Waals equation of state for a classical monoatomic gas is given by

$p = \frac{nT}{1 - b n} - a n^2,$

where $n=N/V$ is the particle number density, $T$ the temperature, $p$ the pressure, $a>0$ quantifies attractive mean-field interactions, and $b>0$ represents excluded-volume repulsion. This form can be derived in the canonical ensemble (CE) or, more generally, formulated for variable particle number in the grand canonical ensemble (GCE) by introducing a shifted chemical potential and solving a transcendental system relating $(p,T,\mu)$ (Vovchenko et al., 2015, Vovchenko et al., 2015).

In reduced variables (normalized to the critical point), the universal form reads: $\pi = \frac{8\tau \omega}{3-\omega} - 3\omega^2,$ with reduced pressure $\pi = p/p_c$, temperature $\tau = T/T_c$, and density $\omega = n/n_c$. The coexistence curve and the critical point can be derived exactly using a parametric representation (Umirzakov, 2018).

Physical interpretation:

• The term $Tn/(1-bn)$ embodies excluded-volume effects (repulsion).
• The negative $a n^2$ term acts as an attractive mean field. At temperatures below the critical point $T_c$, the equation of state yields an S-shaped isotherm and admits coexistence of two macroscopic phases (liquid and vapor), resolved by the Maxwell equal-area construction.

2. Critical Point, Phase Transition, and Fluctuations

The vdW critical point is located by imposing: $$\left. \frac{\partial p}{\partial n} \right|_{T_c} = \left. \frac{\partial^2 p}{\partial n^2} \right|_{T_c} = 0,$$ yielding $T_c = 8a/(27b)$, $n_c = 1/(3b)$, $p_c = a/(27b^2)$. The dimensionless compressibility factor at criticality is universally $z_c = 3/8$ in the classical vdW mean-field model (Umirzakov, 2018, Prodanov, 2022).

The GCE formulation enables the calculation of particle-number fluctuations and higher cumulants:

• Scaled variance $\omega[N]$, skewness $S\sigma$, and kurtosis $\kappa \sigma^2$ are linked to the $n$-th order susceptibilities of $p(T,\mu)$.
• All diverge at the critical point: $$\omega[N] \sim (\tau + \frac{3}{4}\rho^2 + \tau\rho)^{-1}$$.
• Skewness distinguishes gas-like ($S\sigma>0$) and liquid-like ($S\sigma<0$) behavior.
• Strongly intensive measures (such as $\Delta[E,N]$, $\Sigma[E,N]$), designed to suppress volume effects, also diverge at the critical point, making them robust probes of critical phenomena (Vovchenko et al., 2015).

Within the coexistence region, the variance of volume fraction fluctuations remains finite for fixed total system size but also diverges at the critical endpoint (Vovchenko et al., 2015).

3. Quantum and Multicomponent Extensions

Fermionic and bosonic quantum statistics can be incorporated by replacing classical expressions with Fermi-Dirac or Bose-Einstein integrals in the GCE structure, leading to models relevant for nuclear and hadronic matter (Vovchenko et al., 2015, Poberezhnyuk et al., 2015): $$p(T,\mu) = p^{\mathrm{id}}(T,\mu^*) - a n^2, \qquad n(T,\mu) = \frac{n^{\mathrm{id}}(T,\mu^*)}{1 + b n^{\mathrm{id}}(T,\mu^*)},$$ where

$\mu^* = \mu - b p + 2 a n - a b n^2.$

Quantum corrections shift the location of the critical point to lower $T_c$ and $n_c$. In the case of asymmetric nuclear matter, the vdW pressure includes distinct attraction parameters for neutron-neutron and neutron-proton pairs and the EOS is augmented with Fermi corrections from the Thomas–Fermi entropy, ensuring the Nernst theorem is satisfied (Sanzhur, 2022). Extension to multicomponent mixtures in the GCE allows for distinct repulsive and attractive parameters for each pair of species and supports description of asymmetric nuclear matter and hadronic resonance gases (Vovchenko et al., 2017, Ridl et al., 2018).

4. Beyond the Classical van der Waals Model: Generalizations and Integrability

A spectrum of van der Waals-like equations of state has been developed to capture additional physics or improve agreement with experimental data:

• Temperature-dependent attraction: $p = R T/(v - b) - a/(T^k v^2)$ generalizes to the Berthelot EOS for $k=1$ and allows for improved modeling of the adiabatic lapse rate in planetary atmospheres (Hernández et al., 7 Aug 2025).
• Integrable extensions: A four-parameter integrable family of EoS unifies classical vdW, Peng–Robinson, and SRK forms, reproduces their critical data, and maps the phase transition into the language of nonlinear conservation laws and shock wave theory (Giglio et al., 2016).
• Anisotropic and porous media systems: Incorporation of scaled-particle theory for non-spherical particles and mean-field attraction accounts for gas–liquid–nematic coexistence effects with explicit dependence on molecular shape, interaction anisotropy, and matrix porosity (Holovko et al., 2015).

Theoretical studies have revealed geometric and algebraic criteria for vdW-like EoS, including the interpretation of the critical point as a parabolic saddle of the thermodynamic surface with zero Gaussian curvature (Yu et al., 2021) and the mathematical classification of real roots of the cubic polynomial form for the molar volume (Prodanov, 2022).

5. Fluctuation Physics, Limitations, and Extensions

The classical vdW-EoS is mean-field in nature and neglects correlations and critical fluctuations beyond leading divergence (mean-field critical exponents). Quantum vdW models cannot safely extend to reproduce third and higher virial coefficients without violating thermodynamic consistency (specifically, the Third Law of thermodynamics). The induced-surface-tension (IST) generalization provides a systematic method to include surface, as well as volume, contributions to the equation of state, while maintaining quantum thermodynamic consistency and reproducing higher virial coefficients as needed (Bugaev et al., 2017).

Theoretical study of the limiting temperature for quantum Bose systems with vdW interactions reveals the existence of a maximal temperature $T_0$: equilibrium ceases to exist for $T>T_0$ due to divergence of the specific heat and fluctuations, behavior analogous to the Hagedorn temperature in hadron-resonance gas models (Poberezhnyuk et al., 2015).

Generalizations to van der Waals-like models in multicomponent or mixture settings enable the simulation of complex phase diagrams with multiple coexisting phases, as can be implemented with lattice Boltzmann methods (Ridl et al., 2018).

6. Applications in Physics and Other Disciplines

The van der Waals-like equation of state is widely used in thermophysical modeling:

• Statistical hadron/nuclear physics: For the description of nuclear matter, asymmetric nuclear matter, and hadron resonance gases, the vdW-EoS provides a baseline equation with physical parameterization tuned to reproduce empirical saturation properties and critical data (Vovchenko et al., 2015, Sanzhur, 2022, Vovchenko et al., 2017).
• Cosmology: vdW-like fluids have been studied as candidate single-component cosmological models: with parameter tuning the EOS can interpolate between inflationary and radiation- or matter-dominated epochs, and support non-eternal inflation without runaway instabilities (Jantsch et al., 2016, Vardiashvili et al., 2017).
• Astrophysics and planetary science: Temperature-dependent vdW-like EOS improve estimation of adiabatic lapse rates in planetary atmospheres such as those of Titan and Venus (Hernández et al., 7 Aug 2025).
• Econophysics and nonphysical systems: The mathematical structure of the vdW-EoS can be mapped to nonphysical contexts, as in economic equilibrium modeling where criticality and phase coexistence analogues emerge in supply–demand systems (Hongler et al., 2024).

7. Physical Validity, Limitations, and Universal Behavior

The classical vdW-EoS offers limited accuracy for real fluids near the critical point (e.g., failing to recover non-mean-field exponents), at high densities (where clustering and collective effects dominate), and in regimes where quantum statistics or strong correlations are essential (Garces, 2018). Nevertheless, its universality in terms of reduced variables (law of corresponding states) ensures qualitative predictive power across many systems. Quantitative accuracy improves when using exact parametric solutions for coexistence curves and with appropriate parameter fitting to critical data (Umirzakov, 2018).

Strongly intensive fluctuation observables, and universal scaling of cumulants near criticality, have direct application in experimental tests for the existence of a QCD critical point in heavy-ion collisions, where the vdW-EoS provides the foundation for interpreting non-Gaussian fluctuation signals (Vovchenko et al., 2015).

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Selected References

• Universal fluctuation measures and singular behavior at the critical point: (Vovchenko et al., 2015)
• GCE formulation for equilibrium and fluctuations: (Vovchenko et al., 2015)
• Quantum and multicomponent generalizations: (Vovchenko et al., 2015, Vovchenko et al., 2017)
• Integrable and multi-parameter generalizations: (Giglio et al., 2016)
• Applications in cosmology: (Jantsch et al., 2016, Vardiashvili et al., 2017, Abdusattar, 5 Jan 2026)
• Temperature-dependent and anisotropic EOS: (Hernández et al., 7 Aug 2025, Holovko et al., 2015)
• Thermodynamic consistency and higher virial extension: (Bugaev et al., 2017)
• Mathematical analysis of cubic root structure and criticality: (Prodanov, 2022)
• Limiting temperature and Bose-Einstein statistics: (Poberezhnyuk et al., 2015)
• Critical curve fitting and PVT benchmark data: (Umirzakov, 2018)
• Dense-matter and hole-based entropy models: (Garces, 2018)