Entropy, Dynamics and Instantaneous Normal Modes in a Random Energy Model

It is shown that the fraction f of imaginary frequency instantaneous normal modes (INM) may be defined and calculated in a random energy model(REM) of liquids. The configurational entropy S and the averaged hopping rate among the states R are also obtained and related to f, with the results R~f and S=a+b*ln(f). The proportionality between R and f is the basis of existing INM theories of diffusion, so the REM further confirms their validity. A link to S opens new avenues for introducing INM into dynamical theories. Liquid 'states' are usually defined by assigning a configuration to the minimum to which it will drain, but the REM naturally treats saddle-barriers on the same footing as minima, which may be a better mapping of the continuum of configurations to discrete states. Requirements of a detailed REM description of liquids are discussed

Energy landscape of a Lennard-Jones liquid: Statistics of stationary points

Molecular dynamics simulations are used to generate an ensemble of saddles of the potential energy of a Lennard-Jones liquid. Classifying all extrema by their potential energy u and number of unstable directions k, a well defined relation k(u) is revealed. The degree of instability of typical stationary points vanishes at a threshold potential energy, which lies above the energy of the lowest glassy minima of the system. The energies of the inherent states, as obtained by the Stillinger-Weber method, approach the threshold energy at a temperature close to the mode-coupling transition temperature Tc.Comment: 4 RevTeX pages, 6 eps figures. Revised versio

Mean-atom-trajectory model for the velocity autocorrelation function of monatomic liquids

We present a model for the motion of an average atom in a liquid or supercooled liquid state and apply it to calculations of the velocity autocorrelation function $Z(t)$ and diffusion coefficient $D$. The model trajectory consists of oscillations at a distribution of frequencies characteristic of the normal modes of a single potential valley, interspersed with position- and velocity-conserving transits to similar adjacent valleys. The resulting predictions for $Z(t)$ and $D$ agree remarkably well with MD simulations of Na at up to almost three times its melting temperature. Two independent processes in the model relax velocity autocorrelations: (a) dephasing due to the presence of many frequency components, which operates at all temperatures but which produces no diffusion, and (b) the transit process, which increases with increasing temperature and which produces diffusion. Because the model provides a single-atom trajectory in real space and time, including transits, it may be used to calculate all single-atom correlation functions.Comment: LaTeX, 8 figs. This is an updated version of cond-mat/0002057 and cond-mat/0002058 combined Minor changes made to coincide with published versio

Saddles in the energy landscape probed by supercooled liquids

We numerically investigate the supercooled dynamics of two simple model liquids exploiting the partition of the multi-dimension configuration space in basins of attraction of the stationary points (inherent saddles) of the potential energy surface. We find that the inherent saddles order and potential energy are well defined functions of the temperature T. Moreover, decreasing T, the saddle order vanishes at the same temperature (T_MCT) where the inverse diffusivity appears to diverge as a power law. This allows a topological interpretation of T_MCT: it marks the transition from a dynamics between basins of saddles (T>T_MCT) to a dynamics between basins of minima (T<T_MCT).Comment: 4 pages, 3 figures, to be published on PR

Instantaneous Normal Mode Analysis of Supercooled Water

We use the instantaneous normal mode approach to provide a description of the local curvature of the potential energy surface of a model for water. We focus on the region of the phase diagram in which the dynamics may be described by the mode-coupling theory. We find, surprisingly, that the diffusion constant depends mainly on the fraction of directions in configuration space connecting different local minima, supporting the conjecture that the dynamics are controlled by the geometric properties of configuration space. Furthermore, we find an unexpected relation between the number of basins accessed in equilibrium and the connectivity between them.Comment: 5 pages, 4 figure

A single saddle model for the beta-relaxation in supercooled liquids

We study the Langevin equation for a single harmonic saddle as an elementary model for the beta-relaxation in supercooled liquids close to Tc. The input of the theory is the spectrum of the eigenvalues of the dominant stationary points at a given temperature. We prove in general the existence of a time-scale t_eps, which is uniquely determined by the spectrum, but is not simply related to the fraction of negative eigenvalues. The mean square displacement develops a plateau of length t_eps, such that a two-step relaxation is obtained if t_eps diverges at Tc. We analyze the specific case of a spectrum with bounded left tail, and show that in this case the mean square displacement has a scaling dependence on time identical to the beta-relaxation regime of Mode Coupling Theory, with power law approach to the plateau and power law divergence of t_eps at Tc.Comment: Revised versio