Functional renormalization group for classical liquids without recourse to hard-core reference systems: A study of three-dimensional Lennard-Jones liquids

TL;DR

The paper addresses the difficulty of achieving thermodynamically consistent, accurate descriptions of classical liquids near critical points without relying on hard-core reference systems. It extends the functional renormalization group (FRG) framework to three-dimensional liquids, deriving a closed flow equation for the dimensionless free-energy functional $\overline{F}_\lambda[\rho]$ and hierarchical equations for cavity distributions $y^{(n)}_\lambda$, which are truncated using the Kirkwood superposition approximation and made computationally feasible through a Legendre expansion and a controlled $\lambda$-evolution of the interaction. Applied to the Lennard-Jones fluid, the method delivers thermodynamic quantities and pair correlations in better agreement with molecular dynamics than standard closures, while maintaining thermodynamic consistency between different routes; it also captures pressure softening near the critical point and the onset of spinodal behavior. The results suggest that FRG provides a robust, non-perturbative framework for classical liquids that can outperform conventional integral equation theories, with potential for extension to realistic solvents and improved treatment of high-density regimes.

Abstract

In our previous work [Phys. Rev. E 104, 014124 (2021)], we developed a method for analyzing classical liquids using the functional renormalization group (FRG) without relying on a hard-core reference system. In this paper, we extend that method to three-dimensional liquids. We describe an efficient approach for performing the spatial integrals that appear in the renormalization group equations, which is essential for realizing numerical calculations in three dimensions. As a demonstration of our method, we present its application to the Lennard-Jones liquid. By calculating thermodynamic quantities and the pair distribution function near the critical point, we find that, compared with integral equation methods, the FRG approach preserves thermodynamic consistency much more effectively and more accurately reproduces the results of molecular dynamics simulations. Moreover, we successfully capture characteristic phase-transition phenomena with FRG, such as the softening of pressure near the critical temperature. Our results suggest that FRG can provide a more accurate framework for describing classical liquids than conventional methods such as integral equation theories.

Figures (7)

• Figure 1: Density dependence of the pressure $P$, excess free energy $\overline{F}^{\mathrm{ex}}/N$, and excess chemical potential $\overline{\psi}^{\mathrm{ex}}/N$ at $T^*=1.4$. For both the FRG and the integral equation methods (HNC, PY, and KH), results obtained via the virial and compressibility routes are presented and labeled as (Virial) and (Comp.), respectively. In addition, for the FRG, the results obtained using Eqs. \ref{['eq: flow F final']} and \ref{['eq: flow psi final']} are also shown and labeled as (Flow). The MD results are obtained via the virial route.
• Figure 2: Pair distribution function results from FRG, integral equation methods (HNC, PY, KH), and MD at $T^*=1.4$, $\rho^*=0.5$. The lower panel shows the differences between the MD result $g^{(2)}_{\mathrm{MD}}(r)$ and those of the other methods.
• Figure 3: Same as Fig. \ref{['fig: dist0.5']}, but for $\rho^*=0.25$.
• Figure 4: Density dependence of the pressure obtained from FRG. The gray (green) shaded region indicates the spinodal (metastable) region as suggested by the results.
• Figure 5: $\lambda$-dependence of the modulus at $T^*=1.4$ and $\rho^*=0.25, 0.5$.