Self-Averaging Fluctuations in the Chaoticity of Simple Fluids

Abstract

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.

Critical fluctuations and slowing down of chaos

“…This scaling function is a system-size independent measure of the finite-time fluctuations in the Lyapunov spectrum, fluctuations caused by the local heterogeneities in phase space sampled by our simulated trajectories. The basic form of this scaling function is similar to that seen for simple liquids 33 and Hamiltonian lattices 45 . Our calculation of , a dynamical invariant in the long time limit, leads directly to a dynamical timescale for λ 1 ( t ) fluctuations, .…”

“…In spatially-extended dynamical systems where it is known, the scaling of fluctuations is homogeneous across the bulk of the spectrum 26,48,49 . Liquids show this behavior, for example, and all the bulk exponents are strongly self-averaging 33 . However, the critical dynamics break this scaling symmetry—a significant fraction of the bulk exponents self-average weakly as shown in Fig.…”

“…At the critical point, though, repulsive intermolecular forces play a subordinate role compared to critical fluctuations. While the van der Waals picture of fluids was only recently shown to extend to the Lyapunov exponents and their fluctuations 33 , it does not hold near the critical point 8 . A steady 7% decrease with temperature and the clear minimum near T c in the first Lyapunov exponent are direct evidence that critical conditions inhibit the effect of repulsive forces on the dynamics, consistent with the breakdown of the van der Waals picture.…”

“…The order of the thermodynamic limit and the time limit in the definition of the Lyapunov exponents can determine whether a dynamics is chaotic or not 45,47 . Because of the subtleties of the time and thermodynamic limit order, fluctuations in the finite-time Lyapunov exponents of both dissipative and conservative dynamical systems have recently been subject to a finite-size scaling analysis 26,33,45 . In all cases reported to date, the self-averaging of the first exponent is distinct from that of the bulk, the set of 3 N −1 positive exponents that exclude the first.…”

Variational principle for scale-free network motifs

“…Self-averaging network properties have relative fluctuations that tend to zero as the network size n tends to infinity. Several physical quantities in for example Ising models, fluid models and properties of the galaxy display non-self-averaging behavior [33,34,35,36,37,38,39]. We consider motif counts N (H) and call N (H) self-averaging when Var (N (H)) /E [N (H)] 2 → 0 as n → ∞.…”