Gauge invariance and hyperforce correlation theory for equilibrium fluid mixtures

Abstract

We formulate gauge invariance for the equilibrium statistical mechanics of classical multi-component systems. Species-resolved phase space shifting constitutes a gauge transformation which we analyze using Noether's theorem and shifting differential operators that encapsulate the gauge invariance. The approach yields exact equilibrium sum rules for general mixtures. Species-resolved gauge correlation functions for the force-force and force-gradient pair correlation structure emerge on the two-body level. Exact 3g-sum rules relate these correlation functions to the spatial Hessian of the partial pair distribution functions. General observables are associated with hyperforce densities that measure the covariance of the given observable with the interparticle, external, and diffusive partial force density observables. Exact hyperforce and Lie algebra sum rules interrelate these correlation functions with each other. The practical accessibility of the framework is demonstrated for binary Lennard-Jones mixtures using both adaptive Brownian dynamics and grand canonical Monte Carlo simulations. Specifically, we investigate the force-force pair correlation structure of the Kob-Andersen bulk liquid and we show results for representative hyperforce correlation functions in Wilding et al.'s symmetrical mixture confined between two asymmetric planar parallel walls.

Figures (2)

• Figure 1: Two-body gauge correlation functions of the Kob-Andersen liquid at reduced temperature $k_BT/\epsilon=1.1$ and scaled partial bulk densities $\rho_1^b\sigma^3=0.591$ and $\rho_2^b\sigma^3=0.253$. Results are shown as a function of the scaled interparticle distance $r/\sigma$ for species ${\alpha\alpha'}=11$ (first column), 12 (second column), 22 (third column), and for the agglomerated quantities (fourth column). Top row: partial pair distribution functions $g_{{\alpha\alpha'}}(r)$. Second row: the parallel component of the force-gradient correlation function $g_{\nabla f\parallel}^{({\alpha\alpha'})}(r)$ agrees numerically with $g_{\alpha\alpha'}(r)\beta \phi_{\alpha\alpha'}"(r)$, cf. Eq. \ref{['EQforceGradientViaPairPotential']}. Third row: the corresponding perpendicular component $g_{\nabla f\perp}^{({\alpha\alpha'})}(r)$ agrees numerically with $g_{\alpha\alpha'}(r)\beta\phi_{\alpha\alpha'}'(r)/r$, cf. Eq. \ref{['EQforceGradientViaPairPotential']}. Fourth row: the parallel component of the force-force correlation function $g_{f\!f\parallel}^{({\alpha\alpha'})}(r)$ agrees numerically with $-g_{\alpha\alpha'}"(r) - g_{{\nabla\!f}\parallel}^{({\alpha\alpha'})}(r)$, as expected from the sum rules \ref{['EQ3gParallel']} (first to third panel) and \ref{['EQgIdentityAgglomeratedParallel']} (last panel). Bottom row: the corresponding perpendicular component $g_{f\!f\perp}^{({\alpha\alpha'})}(r)$ agrees numerically with $-g_{\alpha\alpha'}'(r)/r - g_{{\nabla\!f}\perp}^{({\alpha\alpha'})}(r)$, as expected from sum rules \ref{['EQ3gPerpendicular']} (first to third panels) and \ref{['EQgIdentityAgglomeratedPerpendicular']} (last panel).
• Figure 2: Specific hyperforce correlation functions of the symmetrical Lennard-Jones mixture of Wilding et al.wilding1997schmid2001wettingwilding2002wilding2003wilding1998koefinger2006eplkoefinger2006jcp at scaled temperature $k_BT/\epsilon=0.98$ in the gas phase at chemical potential $\mu/\epsilon=-3.05$ (left column), in the mixed liquid phase at $\mu/\epsilon=-2.85$ (middle column), and in the demixed liquid phase at $\mu/\epsilon=-2.65$ (right column). The system is asymmetrically confined between an attractive wall (left) and a purely repulsive wall (right) and it is translationally invariant in the $x$ and $y$ directions. Top row: the total density profile $\rho({\mathbf r})=\rho_1({\mathbf r})+\rho_2({\mathbf r})$ is shown as a function of the scaled distance $z/\sigma$ across the pore. Middle row: the gradient of the density profile, $\nabla\rho({\mathbf r}) = \partial \rho(z)/\partial z$, coincides numerically with the $z$-component of $\langle \hat{\beta}\hat{{\bf F}}_{\rm ext}^\circ\hat{\rho}({\mathbf r})\rangle$, according to Eq. \ref{['EQmixGlobalDensity2']}, and with the $z$-component of $\sum_\alpha \langle \beta {\bf F}_U^{\circ(\alpha)}\hat{\rho}_\alpha\rangle$, according to Eq. \ref{['EQmixGlobalDensity3']}. Bottom row: the $zz$-component of the gradient of the agglomerated local interparticle force density, $\nabla \beta {\bf F}_{\rm int}({\mathbf r})$, coincides numerically with the $zz$-component of the correlation function $\langle \beta \hat{{\bf F}}_{\rm ext}^\circ\hat{{\bf F}}_{\rm int}({\mathbf r})\rangle$, according to Eq. \ref{['EQglobalSumRuleInterparticleForce1']}, and with the $zz$-component of $\sum_\alpha\langle \beta \hat{{\bf F}}^{\circ(\alpha)}_{U} \beta\hat{{\bf F}}_{{\rm int}}^{(\alpha)}({\mathbf r}) \rangle - \sum_\alpha\langle \sum_{ij\in{\cal N}_\alpha} \delta({\mathbf r}-{\mathbf r}_i)\nabla_i\nabla_j \beta u({\mathbf r}^N)\rangle$, according to Eq. \ref{['EQglobalSumRuleInterparticleForce2']}. The bottom row displays corresponding simulation snashopts of the gas (left), mixed liquid (middle), and demixed liquid (right) states.