Closed resummed critical-to-classical crossover functions along the critical isotherm of simple fluids, multi-confluent resummation, and the unobserved Gaussian-to-Wilson-Fisher crossover

Near the liquid-gas critical point of a simple fluid, critical behavior crosses over from Ising-like fluctuation-dominated scaling asymptotically close to criticality to mean-field (classical, analytic) behavior at finite distance. Closed crossover functions that resum the infinite Wegner-like corrections-to-scaling expansion exist only along the critical isochore (the Garrabos-Bervillier massive-renormalization mean crossover functions); along the critical isotherm they are still lacking. The problem is to complete the critical-to-classical crossover theory of the {1f} subclass of simple fluids (one-component fluids with short-range molecular interactions): construct explicit closed-form, fully resummed crossover functions along the critical isotherm with a well-defined Ising-like asymptotic limit integrating the best universal exponent and amplitude estimates; extend the resummation beyond the single lowest confluent exponent so that models with an additional irrelevant field remain theoretically coherent; and establish, experimentally or numerically, a complete renormalization-group crossover from the Gaussian fixed point to the Wilson-Fisher fixed point in a real simple fluid.

Background

Setting. A simple fluid — one-component fluid with short-range molecular interactions, the '{1f}' subclass of critical phenomena — has a liquid-gas critical point at temperature $T_c$ and density $\rho_c$. Two critical paths matter. Along the critical isochore ($\rho = \rho_c$, $T \to T_c$ from above) the ordering field is zero and temperature alone drives the departure from criticality. Along the critical isotherm ($T = T_c$) the ordering field $h$ (the chemical-potential-like field conjugate to the density deviation) drives the departure as $h \to 0$.

Two critical regimes and the crossover. Asymptotically close to $T_c$, fluctuations dominate and the fluid belongs to the three-dimensional Ising universality class — the universality class of systems with a scalar order parameter and short-range interactions, named after the Ising lattice model — with critical exponents $\beta \approx 0.326$ (order parameter), $\gamma \approx 1.237$ (susceptibility), $\delta \approx 4.80$ (critical isotherm), $\nu \approx 0.630$ (correlation length). Farther from $T_c$, beyond a fluid-dependent crossover scale, the behavior is mean-field (classical, analytic) with $\beta = 1/2$, $\gamma = 1$, $\delta = 3$, as in van der Waals-type equations of state. The crossover is governed by the Ginzburg number $Gi$ — a non-universal, fluid-dependent measure of the width of the fluctuation-dominated region — or equivalently by $\vartheta^{-1}$, the inverse of the fluid-dependent Ising-like temperature scale factor $\vartheta$ that sets the reduced-temperature width of the asymptotic Ising-like region. This asymptotic region is the preasymptotic domain (PAD), typically narrow — of order $|\tau| \lesssim 10^{-3}$ in the reduced temperature $\tau = (T - T_c)/T_c$, or less, for simple fluids.

Renormalization-group picture. In renormalization-group (RG) language — the framework analyzing critical behavior through the flow of effective Hamiltonians under coarse-graining — mean-field behavior is controlled by the Gaussian fixed point (the scale-invariant fixed point of the RG flow with zero interaction, where fluctuations are negligible) and Ising behavior by the Wilson-Fisher fixed point (the nontrivial interacting fixed point of the three-dimensional scalar theory); the crossover is the flow along the renormalized trajectory connecting them. According to the source work (2016), a physical reference exhibiting a completed theoretical critical-to-classical crossover from the Gaussian fixed point to the non-trivial (Wilson-Fisher) fixed point has never been observed in the subclass of simple fluids — the mean-field approximation is plausibly meaningful for such fluids only near the triple point, far from criticality.

Wegner expansions and resummation. Corrections to the asymptotic Ising scaling laws form the Wegner expansion: an infinite series in powers of $|\tau|^{\Delta}$ with $\Delta = \omega\nu$ ($\omega \approx 0.83$ is the leading confluent correction-to-scaling exponent), plus terms governed by subleading confluent exponents ($\omega_2 \approx 1.8$, etc.). A crossover function is closed and resummed when it encodes the entire infinite correction series, interpolating between the exact Ising asymptotics and the mean-field limit, rather than truncating at finite order. The benchmark is the set of mean crossover functions of Garrabos and Bervillier (2006), computed within the massive renormalization (MR) field-theoretic scheme of Bagnuls and Bervillier — a renormalization scheme defined at nonzero field mass (finite correlation length), in which crossover functions are obtained as resummed expressions — but these exist only along the critical isochore.

Universality structure. Two-scale-factor universality states that the singular thermodynamics near criticality is fixed by two amplitudes. For the {1f} fluid subclass the source work establishes a three-scale-factor set $\{\vartheta, \psi_\rho, g_0\}$ (temperature scale factor, order-parameter scale factor, and a coupling-like parameter) with a universal master length $L^{\{1f\}} = g_0\,\alpha_c = 25.585$; within the PAD, $\vartheta$ and $\psi_\rho$ alone suffice.

Model landscape. The 2016 source paper benchmarks the main crossover model families against xenon data: MR (massive renormalization scheme), MSR (minimal-subtraction renormalization, Schloms-Dohm), CLM (crossover Landau model, Chen-Albright-Sengers), CPM (crossover parametric model, Agayan-Anisimov-Sengers), TGN (numerical crossover from equivalent-neighbor Ising lattice calculations, Luijten-Binder-Meyer), and CMM (crossover master model of Garrabos et al.). All agree on the Ising-like universal features within the PAD, with small persistent numerical differences, and all resum only the single lowest confluent-exponent family.

Problem Statement

The overall scientific goal is to complete the theoretical critical-to-classical crossover description of the {1f} subclass of simple fluids — one-component fluids with short-range molecular interactions near their liquid-gas critical point — whose closed, resummed crossover functions currently exist only along the critical isochore. Three interrelated open threads define this problem. A complete solution must deliver all three threads — (A), (B) and (C). The threads are not independent: (B) extends the construction of (A) and, because it covers both critical paths, contains it as a partial case, so a solution that fully delivers (B) also delivers (A); and (C) is adjudicated against the crossover predictions supplied by (A) and (B). Delivering only a proper subset leaves the remaining thread(s) open and does not resolve the task.

(A) Closed resummed crossover functions along the critical isotherm. Construct explicit closed-form crossover functions for the thermodynamic properties — order-parameter density, susceptibility (compressibility), specific heat — and the correlation length, valid along the critical isotherm $T = T_c$, $h \to 0$, where $h$ is the field conjugate to the order parameter. These must be the critical-isotherm analogues of the Garrabos-Bervillier (2006) mean crossover functions along the critical isochore and must satisfy: (a) the infinite Wegner-like corrections-to-scaling expansion along this critical path is resummed — the functions encode the full correction series, not a finite-order truncation; (b) the asymptotic Ising-like limit is well defined and exactly integrates the universal critical exponents and universal amplitude combinations of the MR Ising-like benchmark (references 1, 2 and 8), including at least first-order confluent corrections; (c) the correct mean-field (classical, analytic) limit is recovered at finite distance from the critical point.

(B) Resummation beyond a single confluent exponent. Extend the crossover-function construction — for both critical paths — so that the contributions of at least a second confluent exponent (subleading correction-to-scaling exponent $\omega_2$, generating corrections $\sim |\tau|^{\omega_2\nu}$ along the critical isochore and governed by the corresponding power of the ordering field $h$ along the critical isotherm) are consistently incorporated, which the source work identifies as required for theoretical coherence as soon as a model includes an additional irrelevant field (an RG coupling that decays under coarse-graining and generates subleading correction-to-scaling terms; for example, a fourth non-universal parameter).

(C) The Gaussian-to-Wilson-Fisher reference crossover. Establish, by precision experiment on a real simple fluid or by controlled numerical simulation mapped to the {1f} subclass, a complete critical-to-classical crossover along the renormalized trajectory — a physical reference in which the renormalization-group flow from the Gaussian fixed point to the Wilson-Fisher fixed point is actually observed. This requires effective critical exponents crossing from their mean-field values to their Ising values over the full predicted range, agreeing within stated uncertainties with the crossover predictions supplied by the constructions of (A) and (B), together with a quantitative account of the crossover scales.

The remaining directions noted in the source — scarcity of high-accuracy measurements inside the PAD, small inter-scheme numerical differences among MR/MSR/CLM/CPM/TGN, and the incomplete mapping between lattice-range-dependent TGN results and fluid microscopic parameters — are context and motivation.

References

1. Bagnuls, C., Bervillier, C. (2002). Classical-to-critical crossovers from field theory. Physical Review E, 65, 066132. DOI 10.1103/PhysRevE.65.066132, https://doi.org/10.1103/PhysRevE.65.066132
2. Garrabos, Y., Bervillier, C. (2006). Mean crossover functions for uniaxial three-dimensional Ising-like systems. Physical Review E, 74, 021113. DOI 10.1103/PhysRevE.74.021113, https://doi.org/10.1103/PhysRevE.74.021113
3. Garrabos, Y., Lecoutre, C., Palencia, F., Le Neindre, B., Erkey, C. (2008). Master crossover functions for one-component fluids. Physical Review E, 77, 021116. DOI 10.1103/PhysRevE.77.021116, https://doi.org/10.1103/PhysRevE.77.021116
4. Anisimov, M. A., Povodyrev, A. A., Kulikov, V. D., Sengers, J. V. (1995). Nature of Crossover between Ising-like and Mean-Field Critical Behavior in Fluids and Fluid Mixtures. Physical Review Letters, 75, 3146-3149. DOI 10.1103/PhysRevLett.75.3146, https://doi.org/10.1103/PhysRevLett.75.3146
5. Luijten, E., Meyer, H. (2000). Crossover behavior in 3He and Xe near their liquid-vapor critical point. Physical Review E, 62, 3257-3261. DOI 10.1103/PhysRevE.62.3257, https://doi.org/10.1103/PhysRevE.62.3257
6. Agayan, V. A., Anisimov, M. A., Sengers, J. V. (2001). Crossover parametric equation of state for Ising-like systems. Physical Review E, 64, 026125. DOI 10.1103/PhysRevE.64.026125, https://doi.org/10.1103/PhysRevE.64.026125
7. Garrabos, Y., Lecoutre, C., Marre, S., Le Neindre, B. (2016). Critical Crossover Functions for Simple Fluids: Non-Analytical Scaling Determination of the Ising-Like Crossover Parameter. Journal of Statistical Physics, 164, 575-615. DOI 10.1007/s10955-016-1554-4, https://doi.org/10.1007/s10955-016-1554-4
8. Garrabos, Y., Lecoutre, C., Marre, S., Le Neindre, B., Hahn, I. (2016). Critical Crossover Functions for Simple Fluids: Towards the Crossover Modelling Uniqueness. Journal of Statistical Physics, 165, 471-506. DOI 10.1007/s10955-016-1627-4, https://doi.org/10.1007/s10955-016-1627-4
9. Garrabos, Y., Lecoutre, C., Marre, S., Hahn, I. (2024). Crossover Master Model of the Equation-of-State for a Simple Fluid: Critical Universality. International Journal of Thermophysics, 45(6), 78. DOI 10.1007/s10765-024-03359-7, https://doi.org/10.1007/s10765-024-03359-7