Perturbed phase-space dynamics of hard-disk fluids

Abstract

The Lyapunov spectrum describes the exponential growth, or decay, of infinitesimal phase-space perturbations. The perturbation associated with the maximum Lyapunov exponent is strongly localized in space, and only a small fraction of all particles contributes to the perturbation growth at any instant of time. This fraction converges to zero in the thermodynamic large-particle-number limit. For hard-disk and hard-sphere systems the perturbations belonging to the small positive and large negative exponents are coherently spread out and form orthogonal periodic structures in space, the "Lyapunov modes". There are two types of mode polarizations, transverse and longitudinal. The transverse modes do not propagate, but the longitudinal modes do with a speed about one third of the sound speed. We characterize the symmetry and the degeneracy of the modes. In the thermodynamic limit the Lyapunov spectrum has a diverging slope near the intersection with the abscissa. No positive lower bound exists for the positive exponents. The mode amplitude scales with the inverse square root of the particle number as expected from the normalization of the perturbation vectors.

Time-dependent mode structure for Lyapunov vectors as a collective movement in quasi-one-dimensional systems

“…Here, values of α ′ , β ′ and T acf (α ′′ and κ) are derived by fitting the auto-correlation function C x (C y ) to Eqs. (19) and (20) (Eq. (21)).…”

“…15(a) is the beginning part of the autocorrelation functions C x , and is given as a linear-log plot to show their exponential decay as straight lines. In this figure the fits to the exponential function (19) with the fitting parameter α ′ are given for the cases (P,P) and (H,P) and the cases (P,H) and (H,H) separately. The dotted line is the fit for the cases (P,P) and (H,P) with the fitting parameter values α ′ ≈ 0.0765, and the broken line is for the cases (P,H) and (H,H) with the fitting parameter values α ′ ≈ 0.0597.…”

Time-Oscillating Lyapunov Modes and the Momentum Autocorrelation Function

“…These steps in the spectrum are accompanied by global wavelike structures in the corresponding Lyapunov vectors, or Lyapunov modes [5,6,7,8]. The significance of this phenomenon is that this structure appears in the vectors associated with the Lyapunov exponents that are closest to zero, therefore it is connected with the slow macroscopic behavior of the system.…”

“…Ref. [7] claims a moving mode structure in δx (n) j as a function of x, in the same Lyapunov steps as the ones having the mode in δy (n) j /p yj , but the relation between these two modes are not known. Another important unsolved problem is the time scale of the oscillation of the Lyapunov mode.…”

Lyapunov modes and time-correlation functions for two-dimensional systems

“…The existence of such a mode structure for Lyapunov vectors was first suggested from the stepwise structure of the Lyapunov spectra for many-hard-core-particle systems [1,2], and led to the discovery of a stationary transverse T-mode structure in the Lyapunov vectors [2,3,4]. Further investigations clarified the other two kinds of Lyapunov modes, the longitudinal L-mode [5] and the momentum proportional P-mode [6,7,8], both of which contain an explicit time-dependence. Combining these three kinds of Lyapunov modes, explained the degeneracies in the stepwise structure of Lyapunov spectra [7,8,9].…”