Scale-Dependent Equation of State

• Scale-Dependent Equation of State is a framework where pressure and energy density depend not only on local state variables but also on the system’s scale, such as size or gravitational field.
• It employs methodologies like finite-volume corrections, scaling fields, and generalized density scaling to capture critical phenomena and phase transitions.
• It is applied in contexts ranging from polymeric liquids and supercooled fluids to gravitational collapse and early-universe cosmology, ensuring analytic consistency across regimes.

A scale-dependent equation of state (EoS) refers to any EoS in which thermodynamic or dynamical variables, such as pressure, energy density, or entropy, depend not only on the local state variables (e.g. density, temperature) but explicitly on the macroscopic or mesoscopic scale (e.g. system size, gravitational potential, or the density field itself), or on a scale-dependent parameter such as a running coupling constant or cluster size. Scale dependence typically arises in situations involving critical phenomena, strong correlations, confinement by external fields, gravitational environments, multifluid or multiphase systems, and in the presence of quantum or finite-size effects. Such EoSs unify regimes otherwise requiring distinct descriptions, capture universal scaling laws, or encode additional physics beyond mean-field theory.

1. Fundamental Concepts and Definitions

A scale-dependent EoS generalizes the classical thermodynamic relation $P=P(\rho,T)$ (pressure in terms of density and temperature), replacing it by

$P = P(\rho,T; \mathcal{S}),$

where $\mathcal{S}$ encodes explicit dependence on scale, which may be system size, density, gravitational field, or a running coupling. Such dependence may enter through:

• Finite-volume corrections: Integrals over phase space, as in Mayer's cluster expansion, depend explicitly on the finite system size $V$ or particle number $N$, leading to $P=P(\rho,T|V)$ (Ushcats et al., 2018).
• Scaling fields: Universal behavior near critical points is encoded in scaling fields, where the EoS is a function of scaling combinations $D$, $\Theta$, etc., themselves scale-dependent (Neumaier, 2014).
• Scaling exponents: The response to intensive or extensive variables may be governed by exponents $\gamma$ that are themselves functions of state or scale, as observed in generalized density scaling (Grzybowski et al., 2013).
• External fields: Gravity or confining potentials can render even the ideal-gas EoS scale-dependent (Kim et al., 2016).
• Running couplings: In quantum-gravitational contexts, Newton's constant may become scale-dependent, affecting the EoS of matter in gravitational collapse (Hassannejad et al., 2024).

2. Scale-Dependent EoS in Classical and Quantum Fluids

2.1 Virial and Cluster Expansions

In standard practice, the pressure of an interacting fluid is expanded in powers of particle density, with coefficients (cluster integrals) assumed volume-independent. However, retaining the true, scale-dependent form of the integrals $b_k(V)$, one obtains

$P = P(\rho,T; \mathcal{S}),$0

where $P = P(\rho,T; \mathcal{S}),$1 and $P = P(\rho,T; \mathcal{S}),$2 attenuates cluster contributions as system density increases. This construction produces a unified, divergence-free description of gas, two-phase coexistence, and dense liquid regimes, with the scale parameter interpolating smoothly between limits (Ushcats et al., 2018).

2.2 Generalized Density Scaling

In supercooled liquids exhibiting strong correlational "isomorph" behavior, dynamic properties and EoS manifest a generalized density scaling form: $P = P(\rho,T; \mathcal{S}),$3 where $P = P(\rho,T; \mathcal{S}),$4 is a density scaling function (often an analytic polynomial or logarithmic in $P = P(\rho,T; \mathcal{S}),$5), and the scaling exponent $P = P(\rho,T; \mathcal{S}),$6 may itself depend on density. This EOS and its configurational pressure analog $P = P(\rho,T; \mathcal{S}),$7 are explicitly scale-dependent via $P = P(\rho,T; \mathcal{S}),$8, and are rigorously compatible with empirical and simulation-based scaling forms for relaxation times and moduli (Grzybowski et al., 2013).

2.3 Thermodynamic Scaling and Murnaghan EoS

For polymeric glass-forming liquids, the Murnaghan EoS

$P = P(\rho,T; \mathcal{S}),$9

characterizes the volume dependence of pressure, where the pressure derivative of the isothermal bulk modulus, $\mathcal{S}$0, physically encodes the material's anharmonicity. Thermodynamic scaling emerges via the invariance of $\mathcal{S}$1 with $\mathcal{S}$2, establishing a deep link between the scaling exponent and intrinsic material properties. Only certain transformed thermodynamic properties (e.g., reduced compressibility) exhibit this collapse, highlighting the subtlety of scale dependence (Douglas et al., 2021).

3. Critical Phenomena and Scaling Fields

Near critical points (e.g., the liquid–vapor critical point in fluids, Ising universality class), classical EoS forms fail to capture non-analyticity and universality. Global analytic EoS formulations employ scaling fields:

• Strong (ordering) field: $\mathcal{S}$3
• Thermal field: $\mathcal{S}$4
• Dependent scaling field: $\mathcal{S}$5

The normal-form analytic EoS reads

$\mathcal{S}$6

where $\mathcal{S}$7 and $\mathcal{S}$8 are universal exponents, and $\mathcal{S}$9 is a universal analytic function. This structure guarantees Ising-type universal scaling near criticality and remains analytic in the one-phase region, resolving previous assumptions about the necessity of singular non-invertible scaling fields. Corrections to scaling and multiphase coexistence (e.g., vapor–liquid–liquid equilibrium) are included via additional analytic scaling correction fields, with the scale dependence embedded in the scaling fields and their analytic structures (Neumaier, 2014).

4. Scale Dependence Induced by External Fields and Gravity

Gravitational or external potential fields introduce explicit scale dependence into the EoS, even for ideal gases. In the presence of uniform gravity over a finite box, the averaged adiabatic EoS becomes: $V$0 where

$V$1

measures the ratio of gravitational energy per unit area to average pressure. In the $V$2 regime (small systems or weak gravity), $V$3 and the classical polytropic law holds; for $V$4, significant corrections arise, relevant near the surfaces of compact stars or in regimes of high curvature (Kim et al., 2016).

5. Scale-Dependent Equations of State in Gravitational Collapse and Modified Gravity

In quantum or scale-dependent gravity models (e.g., asymptotic safety, running Newton's constant), matter collapse yields a scale-dependent EoS, deriving from the junction of an interior FLRW metric to an exterior scale-dependent black hole: $V$5 where all quantities are scaled by the gravitational mass. For vanishing scale dependence ($V$6), the classical Oppenheimer-Snyder dust EoS ($V$7) is recovered; otherwise, effective pressure arises from the gravitational scale dynamics, modifying horizon formation, singularity approach, and energy conditions inside the collapsing body (Hassannejad et al., 2024).

6. Dynamical and Cosmological Implications: Rapidly Evolving EoS Parameters

In early universe cosmology, specifically ekpyrotic and bouncing scenarios, a rapidly evolving EoS parameter $V$8, or equivalently $V$9, causes the background dynamics to imprint a scale-dependent structure on primordial perturbations. During certain phases, the curvature perturbation spectrum is rendered scale-invariant; but the non-Gaussianity at higher order can itself be scale-dependent, growing with wavenumber and leading to breakdown of perturbation theory unless an explicit transition in the EoS is arranged to suppress small-scale power. The imposed scale dependence of $N$0 thus directly governs the range and character of observable scales (Khoury et al., 2011).

7. Practical Regimes and Applications

Scale-dependent EoS formulations find essential application in:

• Describing unified crossover from gas to liquid in finite systems without heuristic Maxwell constructions (Ushcats et al., 2018).
• Capturing the correct scaling of relaxation times or mechanical moduli in glass-forming or supercooled liquids under wide-ranging thermodynamic conditions (Grzybowski et al., 2013, Douglas et al., 2021).
• Modeling critical and multicritical phenomena in fluids and magnets, both for universal scaling and for realistic coexistence and mixture phenomena (Neumaier, 2014).
• Quantifying departures from local thermodynamic equilibrium in gravitationally bound or finite systems, and providing corrections to the EoS in astrophysical objects and quantum gravity contexts (Kim et al., 2016, Hassannejad et al., 2024).
• Determining consistency of dynamical spacetime models with perturbative and non-Gaussian observables in early universe cosmology (Khoury et al., 2011).

In all cases, the explicit introduction of scale dependence into the EoS is essential for unifying regimes, ensuring analytic consistency, or capturing otherwise inaccessible physical behavior across critical or strongly interacting transitions.