Statistical Equilibrium Relations

• Statistical equilibrium relations are mathematical statements that define balance by relating observables, conserved quantities, and transition rates in systems under equilibrium constraints.
• They are applied across contexts—from hydroelastic wave turbulence yielding Rayleigh–Jeans spectra to geometric derivations of Boltzmann–Gibbs and Maxwellian distributions using additive constraints.
• These relations underpin rate-balance models in chemical kinetics and stellar atmospheres, providing insights into steady-state behavior and transient invariants in nonequilibrium evolution.

Searching arXiv for the main paper and related equilibrium/statistical-mechanics papers to ground the article. arXiv search queries:

1. (Vernet et al., 3 Jun 2025)
2. "statistical equilibrium relations"
3. "equilibrium distributions open closed statistical systems" Statistical equilibrium relations are mathematical statements that characterize stationary distributions, spectral laws, fixed points, or thermodynamic conjugacies in systems whose macroscopic or modal statistics are governed by equilibrium constraints. In the literature represented here, the term spans several distinct but structurally related settings: equipartition and Rayleigh–Jeans spectra in truncated or scale-separated wave systems, Boltzmann–Gibbs and Maxwellian marginals derived from equiprobability in high-dimensional phase space, fixed-point balance laws in finite-dimensional stochastic models, exact concentration-ratio invariants in reversible first-order chemical kinetics, and rate-balance equations for atomic populations in stellar atmospheres. Across these contexts, the common content is that equilibrium is encoded by relations among observables, conserved quantities, or transition rates rather than by any single formalism (Vernet et al., 3 Jun 2025).

1. Statistical equilibrium as a flux-free or constrained stationary state

In "Thermodynamics and Statistical Equilibrium of Large-Scale Hydroelastic Wave Turbulence" (Vernet et al., 3 Jun 2025), statistical equilibrium denotes a stationary, flux-free, equipartitioned state of the large-scale Fourier modes $k<k_p$, maintained while the overall system is still externally forced and dissipative. The paper is explicit that this is not equilibrium of the whole apparatus, and not a decay state. It is a partial equilibrium over a subset of scales. The equilibrium sector is the large-scale interval from the first eigenmode $f_L$ up to the forcing onset $f_p$, where spectra are Rayleigh–Jeans-like and the flux vanishes, whereas the nonequilibrium sector is the forcing and dissipation dominated region, especially at $f>f_p$, where the spectrum is steep, approximately $f^{-8}$ (Vernet et al., 3 Jun 2025).

The same structural idea appears in several other settings, but with different state variables. In "Dynamics of ternary statistical experiments with equilibrium state" (Bertotti et al., 2020), equilibrium is the normalized fixed point $\rho$ satisfying $\widehat{\mathbb V}\rho=0$ together with $\rho_0+\rho_1+\rho_2=1$. In "Statistical equilibrium equations for trace elements in stellar atmospheres" (Kubat, 2010), statistical equilibrium means that level populations are determined by balancing all populating and depopulating processes rather than by imposing Saha-Boltzmann equilibrium. In "Coupling and thermal equilibrium in general-covariant systems" (Chirco et al., 2013), equilibrium is characterized by vanishing information flux and equality of local thermal times. These formulations differ in ontology—modes, probabilities, atomic levels, or constrained Hamiltonian subsystems—but each treats equilibrium as a balance relation on a reduced set of variables rather than as a statement that all microscopic dynamics has ceased.

This suggests a useful unifying description: statistical equilibrium relations specify how a subsystem, sector, or marginal distribution behaves when accessible exchanges are balanced and the relevant fluxes vanish or are exactly constrained. A plausible implication is that the mathematical form of equilibrium depends less on the specific substrate than on which degrees of freedom remain dynamically active after conservation laws, truncations, or timescale separations are imposed.

2. Spectral and thermodynamic relations in hydroelastic wave turbulence

The hydroelastic-wave experiment of (Vernet et al., 3 Jun 2025) provides a particularly explicit set of equilibrium relations. The theoretical framework is weak wave turbulence/statistical mechanics of truncated wave systems, and the equilibrium state corresponds to equipartition of kinetic energy among Fourier modes $\mathbf{k}$. For two-dimensional waves, the associated Rayleigh–Jeans energy spectrum is

$$E^{\mathrm{Eq}}(k)=\frac{k_B \theta}{2\pi \rho}\,k,$$

where $f_L$0 is the Boltzmann constant, $f_L$1 is an effective temperature, and $f_L$2 is the fluid density (Vernet et al., 3 Jun 2025).

Using the tensional branch of the dispersion relation,

$f_L$3

the paper derives the equilibrium prediction for the spatial power spectral density of the wave elevation,

$f_L$4

and the temporal spectrum,

$f_L$5

Thus statistical equilibrium predicts both a spatial spectrum $f_L$6 and a frequency spectrum $f_L$7, each with a prefactor proportional to the effective temperature $f_L$8 and inversely proportional to the imposed tension $f_L$9 (Vernet et al., 3 Jun 2025).

The empirical validation is equally central. The large-scale spectra agree with the Rayleigh–Jeans predictions over more than a decade; $f_p$0 from the forcing band down toward the box scale, and LDV measurements show $f_p$1 over more than one decade, from the forcing frequencies $f_p$2 down to the first eigenmode $f_p$3 Hz. The cumulative flux $f_p$4, estimated using the method of Deike et al. based on spectral energy dissipation, is zero in the equilibrium range within experimental resolution when the forcing is sufficiently far away in scale. The paper further reports that measured large-scale dissipation is less than $f_p$5, so it does not spoil the equilibrium regime (Vernet et al., 3 Jun 2025).

The thermodynamic extension of these spectral relations is unusually direct. Integrating the equilibrium temporal spectrum gives

$f_p$6

so $f_p$7. Experimentally, $f_p$8 scales roughly as $f_p$9 at fixed forcing amplitude, yielding $f>f_p$0. The paper then defines entropy through

$f>f_p$1

with $f>f_p$2, and heat capacity via

$f>f_p$3

Under equilibrium, $f>f_p$4, and the heat capacity is temperature independent: $f>f_p$5 Since $f>f_p$6, the paper estimates $f>f_p$7 (Vernet et al., 3 Jun 2025).

The physical interpretation offered there is narrow but consequential: a driven turbulent system can contain a subset of scales accurately described by equilibrium statistical mechanics, provided there is sufficient scale separation, weak large-scale dissipation, and no sustained flux through that band. Because hydroelastic waves admit three-wave interactions and wave action is not conserved, there is no inverse cascade of action toward large scales; large scales are then expected to thermalize rather than carry a persistent upscale flux (Vernet et al., 3 Jun 2025).

3. Marginal equilibrium laws from geometry and equiprobability

A different but classically recognisable class of statistical equilibrium relations is developed in "Equilibrium Distributions in Open and Closed Statistical Systems" (Lopez-Ruiz et al., 2010). There the governing assumption is equiprobability over the accessible phase-space region, either the volume of an allowed body for open systems or the surface of that body for closed systems. The main result is that, for homogeneous systems in the thermodynamic limit $f>f_p$8, open and closed formulations yield the same asymptotic one-component equilibrium laws (Lopez-Ruiz et al., 2010).

For a linear additive constraint,

$f>f_p$9

the marginal equilibrium distribution tends asymptotically to the Boltzmann–Gibbs exponential law

$f^{-8}$0

For a quadratic additive constraint,

$f^{-8}$1

the marginal tends to the Maxwellian Gaussian

$f^{-8}$2

The equivalence of open and closed formulations is asymptotic: finite-$f^{-8}$3 marginals are polynomial cutoff laws whose exponents differ by terms such as $f^{-8}$4 versus $f^{-8}$5, or $f^{-8}$6 versus $f^{-8}$7, but these differences disappear in the thermodynamic limit (Lopez-Ruiz et al., 2010).

This geometric formulation provides a complementary perspective on statistical equilibrium relations. Instead of beginning with a Hamiltonian or a kinetic equation, it derives the equilibrium law of a component by slicing a high-dimensional constrained region and normalizing the induced measure. A plausible implication is that many familiar equilibrium distributions can be understood as properties of concentration of measure under additive constraints, with ensemble equivalence appearing as a consequence of high dimensionality rather than as an independent axiom.

4. Rate-balance and fixed-point relations

Several papers in the source material define statistical equilibrium through explicit balance equations. In the ternary discrete-time model of (Bertotti et al., 2020), the probabilities of three mutually exclusive alternatives satisfy the normalization condition

$f^{-8}$8

and the dynamics is

$f^{-8}$9

with

$\rho$0

The equilibrium state $\rho$1 is defined by

$\rho$2

Under the condition

$\rho$3

the paper gives the explicit equilibrium formulas

$\rho$4

equivalently

$\rho$5

Fluctuations around equilibrium satisfy

$\rho$6

The sign of $\rho$7 and admissibility of $\rho$8 classify the asymptotic scenarios: attraction to $\rho$9, repulsion from $\widehat{\mathbb V}\rho=0$0, convergence to $\widehat{\mathbb V}\rho=0$1, or convergence to $\widehat{\mathbb V}\rho=0$2 (Bertotti et al., 2020).

In stellar-atmosphere NLTE theory, the analogous balance law is level-by-level rather than alternative-by-alternative. The statistical equilibrium equations for level populations $\widehat{\mathbb V}\rho=0$3 are

$\widehat{\mathbb V}\rho=0$4

with $\widehat{\mathbb V}\rho=0$5 and $\widehat{\mathbb V}\rho=0$6 the radiative and collisional transition probabilities per unit time (Kubat, 2010). Here equilibrium is not given by a closed-form distribution such as Boltzmann or Rayleigh–Jeans; it is the zero of a coupled rate-balance operator. The distinction between detailed balance and statistical equilibrium is explicit: in thermodynamic equilibrium, detailed balance holds pairwise for every process and its reverse, whereas in general SE/NLTE only the sum over all rates for each level vanishes (Kubat, 2010).

These examples show that “statistical equilibrium relation” often means a fixed-point or zero-net-rate condition. Spectral equipartition laws and marginal distributions are special cases that arise when those balance conditions can be solved in closed form or reduced to universal asymptotics.

5. Equilibrium information encoded in nonequilibrium evolution

A recurring theme in the data is that equilibrium constants or equilibrium-like profiles can constrain transient or driven dynamics much more strongly than a purely asymptotic reading would suggest. In "Equilibrium relationships for non-equilibrium chemical dependencies" (Yablonsky et al., 2010), reversible first-order reaction systems exhibit exact cross-trajectory ratios that equal equilibrium constants at every time. For the reversible pair

$\widehat{\mathbb V}\rho=0$7

the solutions satisfy

$\widehat{\mathbb V}\rho=0$8

The same pattern persists for $\widehat{\mathbb V}\rho=0$9, three-member reversible cycles, four-member cycles, and more general reversible first-order networks under Onsager/detailed-balance relations. In larger networks the invariant ratio may equal a product of local equilibrium constants along a reversible path, as in

$\rho_0+\rho_1+\rho_2=1$0

for the four-member cycle (Yablonsky et al., 2010).

A different nonequilibrium generalization appears in "Non-Thermal Einstein Relations" (Guichardaz et al., 2016). For a particle obeying

$\rho_0+\rho_1+\rho_2=1$1

with reflecting-wall confinement, translational invariance implies a stationary exponential profile

$\rho_0+\rho_1+\rho_2=1$2

far from the wall. The exact condition determining $\rho_0+\rho_1+\rho_2=1$3 is not, in general, Einstein’s $\rho_0+\rho_1+\rho_2=1$4, but

$\rho_0+\rho_1+\rho_2=1$5

Expanding $\rho_0+\rho_1+\rho_2=1$6 in cumulants shows that the classical Einstein relation is recovered only when higher cumulants vanish, that is, when the process is Gaussian. The telegraph-noise model gives an explicit counterexample in which the exact stationary exponent differs from $\rho_0+\rho_1+\rho_2=1$7 (Guichardaz et al., 2016).

These results qualify a common misconception. Equilibrium relations are not always merely end-state statements. In some linear or constrained settings they appear as exact transient symmetries, while in driven stochastic systems they may survive only in generalized large-deviation form. What remains stable across these cases is the role of detailed balance, conserved structure, or cumulant-generating functions in fixing the relation.

6. Scope, limitations, and interpretive boundaries

The papers collectively draw sharp boundaries around when equilibrium relations hold and what they do not imply. In the hydroelastic case, the equilibrium range is finite and bounded by the lowest mode $\rho_0+\rho_1+\rho_2=1$8 and the forcing cutoff $\rho_0+\rho_1+\rho_2=1$9; finite-size effects are intrinsic, isotropy is only approximate, and the effective temperature depends on forcing amplitude and system parameters. What is universal there is not the value of $\mathbf{k}$0 but the form of the relations $\mathbf{k}$1, $\mathbf{k}$2, zero net flux, $\mathbf{k}$3, and constant heat capacity in the equilibrium band (Vernet et al., 3 Jun 2025).

In the geometric derivation of exponential and Gaussian marginals, the whole construction depends on equiprobability, homogeneity, additive constraints, and the thermodynamic limit. The equivalence of open and closed systems is asymptotic, not exact at finite $\mathbf{k}$4 (Lopez-Ruiz et al., 2010). In the ternary model, admissibility requires $\mathbf{k}$5 and $\mathbf{k}$6; the regression equations alone do not guarantee these inequalities (Bertotti et al., 2020). In chemical kinetics, the exact transient invariants are proved generally only for reversible first-order networks; the nonlinear mass-action examples are special cases only (Yablonsky et al., 2010). In non-thermal sedimentation, if the scaled cumulant generating function $\mathbf{k}$7 does not exist, not only Einstein’s relation but even the exponential stationary form can fail (Guichardaz et al., 2016).

The broader interpretive lesson is that statistical equilibrium relations are exact only relative to a specified set of variables, constraints, and asymptotic regimes. This suggests that debates over whether a system is “in equilibrium” are often less informative than identifying which sector is flux-free, which conserved quantities are active, and which statistical object—state probability, path measure, mode covariance, or rate balance—is being constrained.