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

Abstract

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.

Critical fluctuations and slowing down of chaos

“…Here, these events are molecular collisions that sample the repulsive part of the intermolecular potential. As the system “practices” phase separating, as Widom put it 5 , fewer of these high-energy collisions occur 32 .…”

“…From simulations of two and three-dimensional liquids, the shape of the spectrum depends on the kinetic temperature and density (Supplementary Fig. 2) 32,35,36 . Figure 1b shows the long-time Lyapunov spectrum at the critical density for temperatures spanning the critical temperature, T c .…”

Self-Averaging Fluctuations in the Chaoticity of Simple Fluids

“…This paradigm forms a basis for the statistical mechanics of equilibrium liquids [8][9][10][11] and, more recently, has been considered for the slow dynamics of supercooled liquids [12][13][14]. However, this view neglects the fluidity of liquid matter that comes from the incessant thermal motion of molecules, motion that, because of these intermolecular forces, is intrinsically chaotic [15]. It remains an open question how the liquid state emerges from the Lyapunov instability and deterministic chaos of molecular dynamics [16] and whether the van der Waals picture mirrors a dynamical perspective, where attractive and repulsive interactions play distinct roles in the emergent chaotic behavior of equilibrium liquids.…”

“…Repulsive forces dominate the decay of Lyapunov exponent fluctuations. Attractive forces not only slow the divergence of trajectories [15], they also diminish the rate γ at which the thermodynamic states of liquids emerge through λ 1 . These results are consistent with the idea that subvolumes of liquids made up of repulsive particles will become independent more quickly with increasing system size than those made of particles that can also attract over a relatively longer range.…”

“…Both the WCA and LJ fluids under N V E conditions have symmetric Lyapunov spectra [15] as a result of the conjugate pairing rule [38,39] and the conservation of phase space volume by Liouville's theorem. Figures 3(a) and 3(c) show this symmetry in the diffusion coefficients of the Lyapunov spectra, {D(λ i )}, for the LJ and WCA liquids.…”