Self-Averaging Fluctuations in the Chaoticity of Simple Fluids

Bulk properties of equilibrium liquids are a manifestation of intermolecular forces. Here, we show how these forces imprint on dynamical fluctuations in the Lyapunov exponents for simple fluids with and without attractive forces. While the bulk of the spectrum is strongly self-averaging, the first Lyapunov exponent self-averages only weakly and at a rate that depends on the length scale of the intermolecular forces; short-range repulsive forces quantitatively dominate longer-range attractive forces, which act as a weak perturbation that slows the convergence to the thermodynamic limit. Regardless of intermolecular forces, the fluctuations in the Kolmogorov-Sinai entropy rate diverge, as one expects for an extensive quantity, and the spontaneous fluctuations of these dynamical observables obey fluctuation-dissipation-like relationships. Together, these results are a representation of the van der Waals picture of fluids and another lens through which we can view the liquid state.

Figure 1

(a) Empirical probability density functions of ${\lambda }_1$ from trajectory ensembles of Weeks-Chandler-Andersen fluids with varying numbers of particles $N=100$, 200, 400 and 800 at the kinetic temperature $T=0.9$ and number density $\rho =0.75$. One inset is a semilog plot of the same data. The other inset shows the potential energy $V(r_{ij})$ for the interaction between Lennard-Jones and Weeks-Chandler-Andersen particles $i$ and $j$ as a function of the interparticle distance $r_{ij}$. (b) Snapshot showing 5 out of the 16 system sizes we simulate with $NVE$ molecular dynamics.

Figure 2

Power law decay in the fluctuations of the finite-time Lyapunov exponent, ${\lambda }_1$, as measured by the diffusion coefficient, $D({\lambda }_1)$, with the number of particles, $N$. Here, ${\lambda }_1$ is the exponent associated with the first Lyapunov vector. Data at three temperatures are shown for the (a) Lennard-Jones and (b) Weeks-Chandler-Andersen fluids. The points are simulation data and the lines are linear fits for systems with $N$ from 100 to 2000 particles.

Figure 3

The diffusion coefficient $D({\lambda }_i)$ as a function of the scaled index $i/12000$ for (a) Lennard-Jones and (c) Weeks-Chandler-Andersen fluids with $N=400–2000$ particles. The scale factor normalizes the abscissa; 12 000 is the number of phase space dimensions for the largest system. These data collapse upon rescaling the diffusion coefficient $D({\lambda }_{i/3N})N^{\gamma }$ as a function of the scaled index $i/3N$ for both the (b) Lennard-Jones and (d) Weeks-Chandler-Andersen fluids. There are 16 systems in all, ranging from 100 to 2000 particles. Insets in (b) and (d) show the scaling of $D({\lambda }_{i/3N})$ with system size $N$ for select scaled indices. The points are simulation data and the lines are linear fits. In all cases, the kinetic temperature is $T=0.9$ and the number density is 0.75.

Figure 4

Fluctuations in the finite-time Kolmogorov-Sinai entropy rate, as measured by the associated diffusion coefficient, $D(h_{\mathrm{KS}})$, diverge with $N$. Data at three temperatures are shown for the (a) Lennard-Jones and (b) Weeks-Chandler-Andersen fluids. The points are simulation data and the lines are linear fits for $N$ from 100 to 2000 particles.