Interpolating between pair-potential systems

TL;DR

This work introduces a practical criterion based on constant-potential-energy hypersurfaces $oldsymbol{ m Omega}$ to predict when different simple-liquid models share the same physics. By interpolating between pair potentials (LJ, WCA, IPL, Yukawa) and between KA binary LJ variants, and by constraining density while adjusting temperature via reduced-force matching, the authors show that structure (via $g(r)$) and dynamics (via MSD and diffusion) are nearly invariant across the interpolation when $\lambda$-dependent energy trajectories display few level crossings. The key finding is that approximately identical $oldsymbol{ m Omega}$ suffices to reproduce similar physics across models, reinforcing quasiuniversality and validating an NVU-based perspective that does not require a fixed reference system. The approach is general and scalable to more complex or molecular liquids, offering a framework to compare disparate models and to rationalize why diverse pair-potentials yield similar macroscopic behavior.

Abstract

This paper studies liquid-model systems with almost identical constant-potential-energy hypersurfaces. We simulated continuous interpolations between such systems, specifically between the Lennard-Jones (LJ), Weeks-Chandler-Andersen (WCA), exponent 12 inverse-power-law (IPL), and Yukawa (YK) pair-potential systems. Structure and dynamics were monitored via the radial distribution function and the time-dependent mean-square displacement, respectively. In terms of the interpolation parameter lambda, we argue that two systems have very similar constant-potential-energy hypersurfaces if the potential energies of configurations rarely cross when plotted as functions of lambda. Such absence of "level crossing" applies to a very good approximation for the LJ to WCA transformation, and it also applies to a quite good approximation for the LJ to IPL and the YK to YK transformations (the latter varies the screening length). In all cases, structure and dynamics are shown to be almost invariant as functions of lambda. The density is kept constant when lambda is varied. Temperature must generally be adjusted with lambda, which is done by an iterative "reduced-force-matching" method with no free parameters. We also apply the interpolation strategy to two versions of the Kob-Andersen (KA) binary LJ system and show that a recently introduced shifted-force-cutoff version of this system has constant-potential-energy hypersurfaces, which are almost identical to those of the original KA system. This result rationalizes the previously established fact that the two KA versions have virtually identical physics.

Figures (13)

• Figure 1: The pair potentials studied. Plotted as functions of the pair distance $r$, the figure shows the Lennard-Jones (LJ), Weeks-Chandler-Andersen (WCA), inverse power-law with exponent 12 (IPL), and Yukawa (YK) pair potentials. The Yukawa potential is shown for the two characteristic lengths, $\sigma=1$ and $\sigma=3$, between which we extrapolate (Fig. \ref{['fig5']}).
• Figure 2: The three simulated pair-potential interpolations, each of which is shown for six values of $\lambda$ (legend of (a)). (a) ${\rm LJ}\to{\rm WCA}$, corresponding to cut-and-shifted LJ systems with a cutoff that decreases from 2.5 ($\lambda=0$, defining the standard LJ system) to $2^{1/6}$ ($\lambda=1$, defining the WCA system); (b) ${\rm LJ}\to{\rm IPL}$; (c) ${\rm YK}\to{\rm YK}$. In all cases the potentials vary significantly so a priori one might expect quite different physics.
• Figure 3: Results for the ${\rm LJ}\to{\rm WCA}$ interpolation. The reference state point $(\rho, T)=(1.0,2.0)$ is where the $\lambda=0$ simulation was carried out, which forms the basis for the $\lambda$-scaling plots in (a) and (b). (a) Per-particle potential energy with changing $\lambda$ plotted for 3 equilibrium configurations. (b) Fluctuations of $U$ around the sample-average for 32 independent configurations, scaled with the standard deviation $\sigma(U)$. For $\lambda < 0.75$ there are virtually no "level crossings", and for $\lambda>0.75$ Eq. (\ref{['eq:crit_eq2']}) is still largely obeyed. (c) Radial distribution functions $g(r)$ for six $\lambda$ values. For each $\lambda$ the temperature was adjusted according to the reduced-force-matching criterion Eq. (\ref{['eq:red_force_mat']}), leading only to minor temperature changes (see below). (d) Reduced-unit mean-square displacement as a function of time with changing $\lambda$.
• Figure 4: Results for ${\rm LJ}\to{\rm IPL}$ with reference state point $(\rho, T)=(1.0,2.0)$. (a) and (b) are analogous to the same panels in Fig. \ref{['fig3']}. (c) and (d) show the variation of structure and dynamics if no temperature adjustment is implemented, in which case the physics is not quite invariant. (e) and (f) show structure and dynamics when the temperature is adjusted according to the reduced-force matching method (Appendix). Here both structure and dynamics are invariant to a notably higher degree.
• Figure 5: Results for ${\rm YK}\to{\rm YK}$ with reference state point $(\rho, T)=(0.5,0.02)$. (a) and (b) are analogous to the same panels in Fig. \ref{['fig3']}. (c) and (d) show the structure and dynamics if no temperature adjustment is carried out. The physics changes a lot with $\lambda$. (e) and (f) show the same with temperature adjusted, in which case structure and dynamics are invariant to a very good approximation.