Universality in Driven and Equilibrium Hard Sphere Liquid Dynamics

We demonstrate that the time evolution of the van Hove dynamical pair correlation function is governed by adiabatic forces that arise from the free energy and by superadiabatic forces that are induced by the flow of the van Hove function. The superadiabatic forces consist of drag, viscous, and structural contributions, as occur in active Brownian particles, in liquids under shear and in lane forming mixtures. For hard sphere liquids, we present a power functional theory that predicts these universal force fields in quantitative agreement with our Brownian dynamics simulation results.

Figure 1

Two-body dynamics of the hard sphere fluid. (a) Illustration of the van Hove dynamical two-body correlation function in a bulk liquid of hard spheres of diameter $\sigma$ at time $t=0$ and at $t>0$ (b). (c)–(g) Results for the dynamical decay of the two-body structure of a bulk liquid of hard spheres at packing fraction 0.35 at times $t=0.1\tau$ (left column), $0.3\tau$ (middle column), and $0.6\tau$ (right column) and as a function of the scaled distance $r/\sigma$. (c) Total ($\rho$), self (${\rho }_{\mathrm{self}}$), distinct (${\rho }_{\mathrm{dist}}$), and differential (${\rho }_{\mathrm{\Delta }}$) parts of the van Hove function, as indicated. The results from BD simulation (symbols) of the time evolution and from MC simulation (lines) of the corresponding adiabatic state coincide on the scale of the plot. (d) Self part of the superadiabatic force density ${\mathbf{F}}_{\sup }^{\mathrm{self}}$ as obtained from BD (symbols) and from PFT (solid line); also shown is the adiabatic self force density ${\mathbf{F}}_{\mathrm{ad}}^{\mathrm{self}}$ from MC simulation (blue symbols) and DFT (blue line). The ideal self force density $-k_{B}T\nabla {\rho }_{\mathrm{self}}$ is shown as a reference. (e) Distinct part of the superadiabatic force density ${\mathbf{F}}_{\sup }^{\mathrm{dist}}$ as obtained from BD and from PFT. (f) Differential superadiabatic (drag) force density ${\mathbf{G}}_{\sup }$ as obtained from BD (symbols) and from PFT (black line), and differential adiabatic force density ${\mathbf{G}}_{\mathrm{ad}}$ as arising from the adiabatic self correction. (g) Species-independent superadiabatic force field ${\mathbf{f}}_{\sup }$ as obtained from BD and from PFT, along with the theoretical viscous (${\mathbf{f}}_{\mathrm{visc}}=-{\rho }^{-1}\delta P_t^{\mathrm{visc}}/\delta \mathbf{v}$) and structural contributions (${\mathbf{f}}_{\text{struc}}=-{\rho }^{-1}\delta P_t^{\text{struc}}/\delta \mathbf{v}$), where ${\mathbf{f}}_{\sup }={\mathbf{f}}_{\mathrm{visc}}+{\mathbf{f}}_{\text{struc}}$. For the sake of clarity, the results in (d)–(g) at $t=0.3\tau$ and $0.6\tau$ are multiplied by factors of 2, 4, 8, or 16 as indicated.