Extensivity and additivity of the Kolmogorov-Sinai entropy for simple fluids

According to the van der Waals picture, attractive and repulsive forces play distinct roles in the structure of simple fluids. Here, we examine their roles in dynamics; specifically, in the degree of deterministic chaos using the Kolmogorov-Sinai (KS) entropy rate and the spectra of Lyapunov exponents. With computer simulations of three-dimensional Lennard-Jones and Weeks-Chandler-Andersen fluids, we find repulsive forces dictate these dynamical properties, with attractive forces reducing the KS entropy at a given thermodynamic state. Regardless of interparticle forces, the maximal Lyapunov exponent is intensive for systems ranging from 200 to 2000 particles. Our finite-size scaling analysis also shows that the KS entropy is both extensive (a linear function of system-size) and additive. Both temperature and density control the “dynamical chemical potential,” the rate of linear growth of the KS entropy with system size. At fixed system-size, both the KS entropy and the largest exponent exhibit a maximum as a function of density. We attribute the maxima to the competition between two effects: as particles are forced to be in closer proximity, there is an enhancement from the sharp curvature of the repulsive potential and a suppression from the diminishing free volume and particle mobility. The extensivity and additivity of the KS entropy and the intensivity of the largest Lyapunov exponent, however, hold over a range of temperatures and densities across the liquid and liquid-vapor coexistence regimes.

Figure 1

(a) Interaction potential between the $i\mathrm{th}$ and $j\mathrm{th}$ particle. (b) Snapshots of 4 out of the 15 system sizes we simulate with $NVE$ molecular dynamics: $N=$ 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1300, 1500, 1700, 1900, and 2000.

Figure 2

(a) Representative data showing the KS entropy is linearly extensive in system size, $N$, for both Lennard-Jones ($\square$) and Weeks-Chandler-Andersen ($\circ$) fluids. Data points with error bars are from simulations at fixed number density, $\rho =0.75$, and a kinetic temperature of approximately $T=0.9$. Linear fits of the data (dashed lines) give $h_{K}S=6.0276N+8.6698$ with correlation coefficient 0.999992 for LJ and $h_{K}S=7.4104N+1.1900$ with correlation coefficient 0.999943 for WCA. The quality of these fits supports the extensivity of the KS entropy over an order of magnitude in $N$. The shallow slope of $h_{K}S\left(N\right)$ for the LJ fluid compared to the purely repulsive WCA fluid is due to the attractive forces. The associated Lyapunov spectra of the (b) LJ and (c) WCA fluids with $N=$ 200, 300, 400, 500, 600, 700, 800, 900, 1000, 1100, 1300, 1500, 1700, 1900, and 2000 particles at $\rho =0.75$ and $T=0.9$. The KS entropy is the area under the positive portion of the spectra. For each spectrum, the symmetry about the horizontal axis is a consequence of the conjugate pairing rule.

Figure 3

(a) The largest Lyapunov exponent, ${\lambda }_{\max }$, and (b) the KS entropy, $h_{K}S$, as a function of temperature $T$ for the LJ fluid at $\rho =0.65$, 0.75, and 0.85. (c) ${\lambda }_{\max }$ and (d) $h_{K}S$ as a function of temperature for the WCA fluid at the same number densities. The number of particles is fixed $N=1000$. Statistical errors are smaller than the symbol size.

Figure 4

Normalized Lyapunov spectra, ${\lambda }_i/{\lambda }_{\max }$, for the (a–c) LJ and (d–f) WCA fluid for a range of kinetic temperatures and densities (a, d) $\rho =0.65$, (b, e) $\rho =0.75$, and (c, f) $\rho =0.85$. The number of particles is fixed, $N=1000$.

Figure 5

The function $2h_{K}S/3N{\lambda }_{\max }$ against temperature $T$ at three different number densities for (a) the LJ fluid and (b) the WCA fluid. For both the cases, $N=1000$.

Figure 6

(a) The KS entropy, $h_{K}S$, as a function of the number of particles, $N$, for the Lennard-Jones fluid at (a) three temperatures ($\rho =0.75$) and (b) three densities ($T=0.9$). Data points with error bars smaller than symbols are from simulations at fixed number density. Dashed lines are linear fits. The function $6N$ (solid black line) is shown for reference.

Figure 7

Normalized Lyapunov spectra, ${\lambda }_i/{\lambda }_{\max }$, for the (a–c) LJ and (d–f) WCA fluid at densities between 0.4 and 0.9 and between 0.4 and 1.0, respectively, and kinetic temperatures (a, d) $T=0.8$, (b, e) $T=0.9$, and (c, f) $T=1.0$. The number of particles is fixed, $N=1000$.

Figure 8

The dependence of (a) ${\lambda }_{\max }$ and (b) $h_{K}S$ on the number density $\rho$ at three temperatures for the LJ fluid with $N=1000$ particles. The dependence of (c) ${\lambda }_{\max }$ and (d) $h_{K}S$ on $\rho$ for the WCA fluid at the same three temperatures and $N=1000$.

Figure 9

Dependence of (a) ${\lambda }_{\max }$ and (b) $h_{K}S$ on the number density, $\rho$, at $\sigma =1.00$, 1.10, and 1.25 for the LJ fluid. Dependence of (c) ${\lambda }_{\max }$ and (d) $h_{K}S$ with $\rho$ at the same $\sigma$ for the WCA fluid. The number of particles is fixed at $N=1000$ and the temperature is $T=0.9$. The density range only includes simulations where the systems are in the fluid phase. For $\sigma$ values 1.1 and 1.25, the $\rho$ values up to which the system remains in the fluid phase are 0.75 and 0.55, respectively, for both LJ and WCA fluids. In these data, the units of $h_{K}S$ and $\rho$ are not reduced.

Figure 10

(a) Average number of repelling particles $〈N_{\mathrm{r}}〉$ and the free volume per particle $V_{\mathrm{free}}/N$ as a function of $\rho$ for the LJ fluid at three different temperatures. (b) Variation of $〈N_{\mathrm{r}}〉$ and $V_{\mathrm{free}}/N$ against $\rho$ for systems with WCA interaction potential at three different temperatures. (c) Variation of average potential energy and $V_{\mathrm{free}}/N$ against $\rho$ for WCA fluid at three different temperatures. The number of particles has been kept fixed at $N=1000$.

Figure 11

Average number of repelling particles $〈N_{\mathrm{r}}〉$ and the free volume per particle $V_{\mathrm{free}}/N$ as a function of $\rho$ at $\sigma =1.00$, 1.10, and 1.25 for the (a) LJ and (b) WCA fluid. The number of particles is $N=1000$ and the kinetic temperature is $T=0.9$.