The global isomorphism approach maps the entire liquid-vapor coexistence curve of a continuum fluid onto the binodal of an Ising-like lattice gas through a projective transformation of state variables, with the temperature scale set by a single parameter zz. Applied to the two-dimensional (2D) hard-core attractive Yukawa fluid (HCAYF), it reproduces simulated binodals for screening parameters λ=1\lambda=1 and 1.81.8 with fitted zz, signals the loss of the stable liquid branch at λ∗≈8\lambda_*\approx8 through the divergence of the Zeno-element density scale, and predicts bulk-to-monolayer critical-temperature ratios of about 3.4 (λ=1\lambda=1) and 2.4 (λ=1.8\lambda=1.8). The problem is whether that correspondence is exact and predictive: (1) establish or refute the mathematical status and uniqueness of the fluid–lattice-gas correspondence, i.e. which Ising-like variant (nearest-neighbor, further-neighbor, which lattice geometry) it selects; (2) derive zz theoretically for the Yukawa potential, which lacks the scaling homogeneity that fixes zz for Mie potentials, in both two and three dimensions; and (3) test the predicted bulk-to-monolayer critical-temperature ratios by independent 3D/2D simulations or monolayer experiments. The three questions are open: the 2025 program review (Bulavin, Kulinskii, Katts and Maslechko, 2025) states that the sketched microscopic derivation 'cannot be considered as a proof'; the only follow-up work provides a mean-field estimate of zz; and the predicted monolayer ratios have not been tested by independent simulation or experiment.

Background

The hard-core attractive Yukawa fluid (HCAYF) is a standard coarse-grained model for systems with screened electrostatic attraction — charged colloids, protein solutions, and dusty plasmas. Its pair potential is infinite for r<σr<\sigma and equals Φ(r)=−ε (r/σ)−1e−λ(r/σ−1)\Phi(r)=-\varepsilon\,(r/\sigma)^{-1}e^{-\lambda(r/\sigma-1)} for r≥σr\ge\sigma, where σ\sigma is the particle diameter, ε\varepsilon the attraction depth, and λ\lambda the inverse screening length: larger λ\lambda means shorter-ranged attraction. Whether stable liquid–vapor coexistence exists at all depends on λ\lambda, because attraction that is too short-ranged cannot sustain a liquid. In three dimensions (3D), simulations find stable liquid–vapor equilibrium for λ≲15\lambda\lesssim15; in two dimensions, relevant to adsorbed monolayers, simulations of a softened Yukawa potential found coexistence stable only for λ<3\lambda<3 and metastable with respect to vapor–solid equilibrium for λ>4\lambda>4 (Méndez-Maldonado, González-Melchor and Alejandre, 2012), while a discrete perturbation theory could treat only λ=1\lambda=1 and 1.81.8 with semiquantitative accuracy (Trejos, Santos and Gámez, 2018). The 2D binodal is hard to obtain by standard liquid-state methods because the 2D Ising critical exponent β=1/8\beta=1/8 makes the coexistence dome very flat.

The global isomorphism approach (Kulinskii, 2010; Bulavin and Kulinskii, 2010) is an analytic alternative built on two empirical regularities of the coexistence region. First, the Zeno line — the locus where the compressibility factor Z=P/(ρT)Z=P/(\rho T) equals one — is nearly straight in the density–temperature plane for many fluids, ρ/ρB+T/TB=1\rho/\rho_B+T/T_B=1, with the Boyle parameters TB,ρBT_B,\rho_B fixed by the second and third virial coefficients B2,B3B_2,B_3 (the coefficients of ρ\rho and ρ2\rho^2 in the low-density expansion of the pressure, P=ρT(1+B2ρ+B3ρ2+⋯ )P=\rho T(1+B_2\rho+B_3\rho^2+\cdots)). Second, the law of rectilinear diameter (LRD) states that the mean of the coexisting liquid and gas densities varies linearly, ρl+ρg=2ρc[1+A(1−T/Tc)]\rho_l+\rho_g=2\rho_c[1+A(1-T/T_c)]. The approach hypothesizes that the LRD of a continuum fluid is the image of the exact particle–hole symmetry of a lattice gas — an Ising model in occupation-number form with Hamiltonian H=−∑⟨ij⟩J ninj−h∑iniH=-\sum_{\langle ij\rangle}J\,n_in_j-h\sum_in_i, ni=0,1n_i=0,1 — and that the entire fluid binodal is obtained from the lattice-gas binodal (in 2D known exactly from Onsager's 1944 solution, x=12(1±f(t)1/8)x=\tfrac{1}{2}\left(1\pm f(t)^{1/8}\right) with f(t)=1−sinh⁡−4(2J/t)f(t)=1-\sinh^{-4}(2J/t)) by the projective transformation

ρ=ρ∗ x1+zt~,T=T∗ zt~1+zt~,z=TcT∗−Tc,\rho=\rho_*\,\frac{x}{1+z\tilde t},\qquad T=T_*\,\frac{z\tilde t}{1+z\tilde t},\qquad z=\frac{T_c}{T_*-T_c},

where xx is the lattice-gas site occupation, t~=t/tc\tilde t=t/t_c the lattice temperature reduced to its critical value, and T∗,ρ∗T_*,\rho_* the intercepts of the Zeno element ρ/ρ∗+T/T∗=1\rho/\rho_*+T/T_*=1 — the line tangent to the liquid branch of the binodal as T→0T\to0. Unlike the virial Boyle parameters, T∗T_* and ρ∗\rho_* are computed from a van der Waals-type coarse-graining, B2vdW(T∗)=0B_2^{\rm vdW}(T_*)=0 and ρ∗=(T∗/B3(T∗)) dB2/dT∣T∗\rho_*=(T_*/B_3(T_*))\,\left.dB_2/dT\right|_{T_*}, so ρ∗\rho_* depends on the third virial coefficient. The one remaining parameter zz fixes the critical point through Tc=T∗z/(1+z)T_c=T_*z/(1+z) and ρc=ρ∗/(2(1+z))\rho_c=\rho_*/(2(1+z)). For Mie (inverse-power) potentials, scaling homogeneity fixes zz; the Yukawa potential has no such homogeneity, so in the 2D application (Katts and Kulinskii, 2023) zz was fitted to the gas branch of simulated binodals, giving z=0.51z=0.51 at λ=1\lambda=1 and z=0.70z=0.70 at λ=1.8\lambda=1.8, with critical temperatures within 1–2% of the 2012 simulation values and critical densities within roughly 5–8%. The same machinery was first developed for the 3D HCAYF (Katts and Kulinskii, 2022), where the Zeno-element parameters were shown to mark the loss of liquid stability while the virial Zeno-line parameters show no anomaly.

Within this framework the third virial coefficient acts as a stability marker: when B3(T∗)→0B_3(T_*)\to0, the density scale ρ∗→∞\rho_*\to\infty, the low-temperature tangent becomes horizontal, and the mapping breaks — interpreted as disappearance of the stable liquid branch. For the 2D HCAYF this occurs at λ∗≈8\lambda_*\approx8 (with the threshold decreasing for softer repulsive cores), while the existing 2D soft-core simulations place instability near λ≈4\lambda\approx4 — a gap the source paper itself judges 'reasonably consistent' and attributes in part to the softer repulsive core used in those simulations; the approach also predicts bulk-to-monolayer critical-temperature ratios T3c/T2c≈3.4T_{3c}/T_{2c}\approx3.4 (λ=1\lambda=1) and ≈2.4\approx2.4 (λ=1.8\lambda=1.8), built on the 3D Yukawa corresponding-states data of Orea and Duda (2008). What is not known — the subject of this problem — is whether the correspondence itself is mathematically exact, which lattice gas it selects, whether zz can be derived rather than fitted, and whether the two sharp 2D quantitative predictions survive independent tests.

Problem Statement

Decide whether the liquid–vapor equilibrium of the hard-core attractive Yukawa fluid (HCAYF) is exactly isomorphic to an Ising-like lattice gas, make the correspondence predictive without fitted parameters, and validate its discriminating two-dimensional predictions. Throughout, the HCAY pair potential is Φ(r)=∞\Phi(r)=\infty for r<σr<\sigma and Φ(r)=−ε(r/σ)−1e−λ(r/σ−1)\Phi(r)=-\varepsilon(r/\sigma)^{-1}e^{-\lambda(r/\sigma-1)} for r≥σr\ge\sigma, where σ\sigma is the particle diameter, ε\varepsilon the attraction depth, and λ\lambda the inverse screening length. The claimed isomorphism is the projective transformation

ρ=ρ∗ x1+z t~,T=T∗ z t~1+z t~,\rho=\rho_*\,\frac{x}{1+z\,\tilde t},\qquad T=T_*\,\frac{z\,\tilde t}{1+z\,\tilde t},

between fluid states (ρ,T)(\rho,T) and lattice-gas states (x,t~)(x,\tilde t), where xx is the lattice-gas site occupation, t~=t/tc\tilde t=t/t_c is the lattice temperature reduced to its critical value, and T∗,ρ∗T_*,\rho_* are the intercepts of the Zeno element ρ/ρ∗+T/T∗=1\rho/\rho_*+T/T_*=1 — the line tangent to the liquid branch of the binodal as T→0T\to0, fixed by the virial coefficients of the HCAY potential (the coefficients BnB_n in the low-density pressure expansion P=ρT(1+B2ρ+B3ρ2+⋯ )P=\rho T(1+B_2\rho+B_3\rho^2+\cdots)) through B2vdW(T∗)=0B_2^{\rm vdW}(T_*)=0 and ρ∗=(T∗/B3(T∗)) dB2/dT∣T∗\rho_*=(T_*/B_3(T_*))\,\left.dB_2/dT\right|_{T_*}. Here the van der Waals coarse-graining splits the potential into the hard core and its attractive tail Φattr(r)=Φ(r)\Phi_{\rm attr}(r)=\Phi(r) for r≥σr\ge\sigma, and fixes T∗=a/bT_*=a/b, with a=−π∫σ∞Φattr(r) r dra=-\pi\int_\sigma^{\infty}\Phi_{\rm attr}(r)\,r\,dr and b=π2σ2b=\tfrac{\pi}{2}\sigma^2 in 2D, and a=−2π∫σ∞Φattr(r) r2 dra=-2\pi\int_\sigma^{\infty}\Phi_{\rm attr}(r)\,r^2\,dr and b=2π3σ3b=\tfrac{2\pi}{3}\sigma^3 in 3D; ρ∗\rho_* then follows from the full virial coefficients through the relation above. The single remaining parameter z=Tc/(T∗−Tc)z=T_c/(T_*-T_c) fixes the critical point through Tc=T∗z/(1+z)T_c=T_*z/(1+z) and ρc=ρ∗/(2(1+z))\rho_c=\rho_*/(2(1+z)). Concretely:

(1) Status and uniqueness. Establish or refute, from microscopic premises (the fluid Hamiltonian with the HCAY potential together with a specified lattice-gas Hamiltonian), that this projective transformation between fluid states (ρ,T)(\rho,T) and lattice-gas states (x,t~)(x,\tilde t) holds exactly over the coexistence region, making explicit the domain over which exactness is claimed — in particular, whether it is asserted to extend into the critical fluctuational region, where the rectilinear-diameter law acquires singular corrections, or to hold only outside that region; determine whether the nearest-neighbor lattice gas is uniquely selected, or whether Ising-like variants with further-neighbor couplings or different lattice geometries define different, equally consistent mappings with different zz.

(2) Parameter determination. Derive zz theoretically — not by fitting to coexistence data (the binodal or the critical points) — for the Yukawa potential, which lacks the scaling homogeneity that fixes zz for Mie (inverse-power) potentials, as a function of λ\lambda in both 2D and 3D; alternatively, establish that zz cannot be determined without fitting to coexistence data.

(3) Cross-dimensional test. Test the predicted bulk-to-monolayer critical-temperature ratios T3c/T2c≈3.4T_{3c}/T_{2c}\approx3.4 at λ=1\lambda=1 and ≈2.4\approx2.4 at λ=1.8\lambda=1.8 by consistent 3D and 2D simulations, or by experiments on monolayers whose effective pair interaction approximates the HCAY form.

All three items are mandatory and independent: the exactness sub-question of item (1) is settled either by a proof or by a refutation, but that outcome does not by itself settle the uniqueness sub-question, which must be resolved separately by determining whether the nearest-neighbor lattice gas is uniquely selected or whether Ising-like variants define equally consistent mappings with different zz; items (2) and (3) must still be addressed on their own terms if exactness is refuted.

The overarching question these items address is whether the fluid–lattice-gas global isomorphism is an exact structural property of the Yukawa fluid, or a flexible fit to binodal data.

References

  1. A. Katts and V. Kulinskii, 'Global isomorphism approach: attractive Yukawa fluid, 2D case', Journal of Molecular Liquids 388, 122736 (2023). DOI: 10.1016/j.molliq.2023.122736; arXiv:2307.12147, https://arxiv.org/abs/2307.12147.
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