Universal Equation of State (UEOS)

• UEOS is a universal analytic relation that links macroscopic thermodynamic variables using first principles, symmetry, and scaling laws.
• It generalizes classical models by deriving state equations from fundamental laws such as the virial theorem and renormalization group scaling, reducing to models like Mie–Grüneisen under specific conditions.
• The framework has broad applications ranging from extreme matter conditions and critical phenomena to machine-learned potentials in materials science, enabling consistent cross-disciplinary insights.

A universal equation of state (UEOS) is a closed analytic relation among macroscopic thermodynamic variables that is not limited to a specific model, material, or regime, but is governed by principles and symmetries that confer wide, sometimes truly universal, applicability. In sharp contrast to empirical or system-specific EOS models, a UEOS is constructed using fundamental laws, such as the first law of thermodynamics, generalized virial theorems, renormalization group scaling, or symmetry arguments, and seeks to encapsulate the essential global structure of state-space for broad classes of physical systems. Modern research has established UEOS frameworks for fluids and solids under extreme conditions (Chefranov, 2022), critical phenomena (Neumaier, 2014), quantum gases both at equilibrium and far from equilibrium (Dogra et al., 2022, Martirosyan et al., 2024), warm dark matter in astrophysics (Vega et al., 2013), and molecular binding energies for materials modeling (Hu et al., 11 Feb 2025). These universal forms provide both predictive analytic relations and a foundation for physical understanding across multiple scales and disciplines.

1. First-Principles Universal Equation of State for Arbitrary Media

Chefranov has formulated a UEOS derived explicitly from the first law of thermodynamics and a generalized virial theorem that permits a variable power-law exponent of the pair interaction potential (Chefranov, 2022). The construction proceeds as follows:

• First law and specific energies: Starting from

$dE = T\,dS - p\,dV,$

and decomposing $E = E_{\rm kin} + E_{\rm pot}$ per mass, the system admits state variables $u$, $u_{\rm kin}$, $u_{\rm pot}$ relating the total, kinetic, and potential specific energies.

• Generalized virial theorem: Assuming the internal potential energy is quasi-homogeneous with a degree $d(p,T)$ leads to the virial-type closure

$$p = \rho\frac{kT}{m} + \frac{\rho}{3}d(p,T)u_{\rm pot}(p,T).$$

• Riemann–Hopf–type equation: Eliminating $p$ between the thermodynamic and virial equations yields a nonlinear PDE for $w = \ln(u_{\rm pot}/u_0)$,

$$\frac{\partial w}{\partial y} + (w+1)\frac{\partial w}{\partial x} - w = 0$$

with $x,y$ logarithms of $T$ and $p$, respectively.

• Closed-form solution and construction of the UEOS: The solution provides a self-consistent, analytic relation for the exponent $d(p,T)$ and, ultimately, the pressure, specific energy, and potential energy as

$$p(\rho,T) = \rho\frac{kT}{m} + \frac{\rho}{3}\,\mathcal{D}(p,T)\,u_0\,\exp\left[\mathcal{D}(p,T)\ln(\rho/\rho_0)\right]$$

$$u(\rho,T) = \frac{3}{2}\frac{kT}{m} + u_0\,\exp\left[\mathcal{D}(p,T)\ln(\rho/\rho_0)\right]$$

where $\mathcal{D}(p,T)$ is determined explicitly by the solution of the constituent PDE.

Comparison with Mie–Grüneisen: The UEOS reduces to the classical Mie–Grüneisen EOS when $d$ is a constant, but uniquely generalizes it by determining $d(p,T)$ from first principles, accommodating arbitrary interparticle potentials and dispensing with ad hoc fitting of Grüneisen parameters, heat capacities, or higher virial expansion terms (Chefranov, 2022).

2. Universal Scaling Laws and Critical Phenomena

Near criticality, the UEOS reflects universality and scaling dictated by renormalization group theory. In the Ising universality class, all thermodynamic systems admit a canonical form (Neumaier, 2014):

$$\Sigma^2 = D^{2e_{-1}} W(D^{-e_0}\Theta, D^{-e_1}I_1, \ldots)$$

where $\Sigma$ (strong scaling field), $\Theta$ (thermal scaling field), and $D$ (dependent scaling field) parametrize the singular part of the free energy, $e_{-1}$ and $e_0$ are universal critical exponents, and $W$ is a universal analytic function. This construction guarantees:

• Correct critical exponents and scaling along coexistence (e.g., liquid–gas) and in the one-phase supercritical region.
• Analyticity and non-singular behavior away from critical points.
• Capability to capture multiphase equilibria, including vapor–liquid–liquid coexistence, beyond what is possible in revised-scaling approaches.

This UEOS formalism is model-independent, requiring only identification of the relevant scaling fields for the system at hand.

3. Universal Equations of State in Quantum Gases

3.1. Unitary Fermi Gases

For spin-balanced two-component fermions at unitarity, the UEOS is constructed via the virial cluster expansion and universal scaling arguments (Bhaduri et al., 2012, Murthy et al., 2013, Silva, 2016). Key forms include:

• All-orders cluster expansion:

$$\Delta b_l = (-1)^l\,\frac{\Delta b_2}{2^{(l-1)(l-2)/2}}$$

for $l\ge2$, where $\Delta b_2$ is the exact two-body cluster shift. The partition function and thermodynamic quantities then follow as closed analytic series, showing excellent agreement with experiment up to $z\sim5$ (homogeneous) or $z\sim10$ (trapped) (Bhaduri et al., 2012).

• Fermi-integral expansion:

$$P(T,\mu) = \alpha (k_B T)^{5/2}\sum_{\nu=\{5/2,3/2,1/2,-1/2\}} c_\nu\,f_\nu(z)$$

with virial coefficients and Bertsch parameter $\xi$ fixing the $c_\nu$, yielding a universal description up to fugacity $z\sim18$ (Silva, 2016).

• Spectral and compressibility effects: The self-consistent $T$-matrix (Luttinger-Ward) and phenomenological Fermi-integral ansätze quantitatively reproduce contact density, pseudogap physics, and thermodynamic response down to the onset of pairing (Bauer et al., 2013).

3.2. Wave Turbulence and Far-From-Equilibrium Universal Laws

In Bose gases driven far from equilibrium, experimental and numerical studies establish a universal relation between the amplitude of the turbulent momentum distribution $n_0$ and the momentum-space energy flux $\varepsilon$. These relations are nontrivial generalizations of equilibrium EoS and take the form:

• Gross–Pitaevskii model (numerical):

$$\frac{n_0}{n} = C\left(\frac{\tau\varepsilon}{n\zeta}\right)^b,$$

with $b\simeq0.67\pm0.02$, $C=29\pm2$, $\tau$ the interaction time scale, and $\zeta=gn$ the mean interaction energy. The observed exponent $2/3$ does not arise in wave-turbulent kinetic theory, signifying genuinely nonperturbative behavior (Martirosyan et al., 2024).

• Cold atom experiment (Bose gas):

$n_0 = A_0\,\Phi(\varepsilon/\varepsilon_0),$

with $A_0$, $\varepsilon_0$ reference scales set by $n$ and $a$ (density and interaction), and $\Phi$ a universal but nontrivial function combining regimes $n_0\propto\varepsilon^{1/3}$ (perturbative) and $n_0\propto\varepsilon^{2/3}$ (strong driving) (Dogra et al., 2022).

These UEOS relations are robust to changes in drive, dissipation, and system history, and admit a quasi-static thermodynamic interpretation for nonequilibrium steady-states.

4. Universal EOS in Materials, Polymers, and Machine Learning Potentials

The Rose et al. universal binding-energy relationship,

$$E^*(r^*) = - (1 + r^*) \exp(-r^*), \quad r^* = a(r/r_0 - 1),$$

encodes the normalized cohesive energy across metals and condensed phases, underpinning a UEOS that manifests as a master curve when plotted in these reduced variables. Recent advances exploit this UEOS in the construction of machine-learning interatomic potentials, notably the SUS2-MLIP model (Hu et al., 11 Feb 2025):

• Physics-informed architecture: The radial part of the interaction potential is constrained by the UEOS scaling law, universally for all element pairs.
• Parameter decoupling: Only two scaling parameters per pair (stiffness $a$ and equilibrium distance $r_0$) are needed for all elements, reducing parameter count and enhancing generalizability.
• Nonlinearity embedding: The universal potential is extended by nonlinear transforms (e.g. Chebyshev polynomials on transformed distances), accessing super-linear expressivity.

These features simultaneously yield highly efficient parameterization, robust transferability across chemical systems, and faithful reproduction of complex physical responses.

5. Universal Equations of State in Polymer Solutions and Gels

The UEOS for linear neutral polymers in good solvent relates the reduced osmotic pressure $\hat{\Pi}$ and reduced concentration $\hat{c}$ via a single universal crossover function (Yasuda et al., 2020):

$\hat{\Pi} = f(\hat{c}),$

with

$$\hat{\Pi} = \frac{\Pi M}{cRT}, \qquad \hat{c} = \frac{c}{c^*}, \qquad c^* = \frac{1}{A_2 M},$$

where $A_2$ is the second virial coefficient. This function interpolates between the dilute virial expansion ($\hat{\Pi}\sim1+\hat{c}$) and semidilute power law regime ($\hat{\Pi} \propto \hat{c}^{1/(3\nu-1)}$ with $\nu\approx0.588$). Through gelation, the same universal function describes the osmotic pressure of sol and gel states, as confirmed experimentally and theoretically.

6. Symmetry-Based Universal EOS for Real Substances

A symmetry-based macroscopic UEOS interpolates between the ideal gas limit ($PV=RT$) and a hypothesized "ideal solid" limit ($TS=R_p P$, $R_p<0$), capturing thermodynamics from vacuum to ultrahigh compressions (Xue et al., 2020):

$$V(P,T) = \frac{R'T}{P} - R_p'\ln\frac{P}{P_0} + C_p'\ln\frac{T}{T_0} + V_0,$$

$$S(P,T) = C_p'\ln\frac{T}{T_0} - R'\ln\frac{P}{P_0} + R_p'\frac{P}{T} + S_0,$$

where $R'$, $R_p'$, $C_p'$, $C_v'$, etc., are material-specific but state-independent. This formalism provides a four-constant analytic EOS valid over broad $(P,T)$ domains, outperforming traditional empirical or virial-based models at both extremes. Thermodynamic consistency and correct limits (ideal gas for $P\to0$, ideal solid for $T\to0$) are manifest, and interpolation is performed at the level of thermodynamic derivatives, ensuring global stability and absence of singularities.

7. Significance, Applications, and Generalization

Universal equations of state enable:

• Predictive modeling: Across single- and multi-phase regimes, critical and far-from-equilibrium states, and diverse physical systems, a UEOS removes the need for ad hoc closures or system-specific fitting.
• Comparison and extension: UEOS construction clarifies the origin and limits of classical EOS (e.g. van der Waals, Mie–Grüneisen), their relation to underlying microscopic symmetries, and generalizes to encompass quantum, relativistic, and nonequilibrium regimes.
• Cross-disciplinary transfer: The same analytic forms or scaling relations appear in astrophysical dark matter halos, machine-learned interatomic potentials, colloidal and polymer solutions, quantum fluids, and highly compressed matter, often with only parameter redefinition required.
• Physical insight: UEOS formalisms expose the hierarchies and saturating behaviors enforced by fundamental symmetries and dimensional analysis (e.g., homogeneity, scaling, universality at criticality), illuminating the deep connections between disparate systems (Chefranov, 2022, Neumaier, 2014, Xue et al., 2020).

The continued refinement and adoption of UEOS formulations is anticipated to further unify statistical thermodynamics across classical, quantum, and driven systems.