We present a theory of the dynamics of monatomic liquids built on two basic ideas: (1) The potential surface of the liquid contains three classes of intersecting nearly-harmonic valleys, one of which (the "random" class) vastly outnumbers the others and all whose members have the same depth and normal mode spectrum; and (2) the motion of particles in the liquid can be decomposed into oscillations in a single many-body valley, and nearly instantaneous inter-valley transitions called transits. We review the thermodynamic data which led to the theory, and we discuss the results of molecular dynamics (MD) simulations of sodium and Lennard-Jones argon which support the theory in more detail. Then we apply the theory to problems in equilibrium and nonequilibrium statistical mechanics, and we compare the results to experimental data and MD simulations. We also discuss our work in comparison with the QNM and INM research programs and suggest directions for future research.

We investigate the distribution of local minima in the potential-energy landscape of metals. The density of energy minima is calculated for Na by using a pair-potential method to quench from stochastic configurations for system sizes ranging from 1 to 4000 atoms. We find a minimum system size, approximately 150 atoms, above which the density of energy minima is dominated by one sharp peak. As the system size is increased, the peak position converges to an asymptotic value and its width converges to zero. The findings of the pairpotential method for Na are confirmed by first-principles calculations of amorphous Al and V. Finally we present an example in which our results are applied to the complex bulk metallic glass Zr 52.5 Cu 17.9 Ni 14.6 Al 10 Ti 5 ͑Vitreloy 105͒. The calculated density and bulk modulus of the Vitreloy are in good agreement with experiments. The analysis presented here shows that the thermodynamic limit is better described by one large supercell calculation than by an average over many smaller supercell calculations. We argue that the minimum cell size that is needed to accurately perform such a large supercell calculation for metallic glasses is about 150 atoms.

It is possible in principle to probe the many-atom potential surface using density functional theory (DFT). This will allow us to apply DFT to the Hamiltonian formulation of atomic motion in monatomic liquids by Wallace [Phys. Rev. E 56, 4179 (1997)]. For a monatomic system, analysis of the potential surface is facilitated by the random and symmetric classification of potential-energy valleys. Since the random valleys are numerically dominant and uniform in their macroscopic potential properties, only a few quenches are necessary to establish these properties. Here we describe an efficient technique for doing this. Quenches are done from easily generated "stochastic" configurations, in which the nuclei are distributed uniformly within a constraint limiting the closeness of approach. For metallic Na with atomic pair potential interactions, it is shown that quenches from stochastic configurations and quenches from equilibrium liquid molecular dynamics configurations produce statistically identical distributions of the structural potential energy. Again for metallic Na, it is shown that DFT quenches from stochastic configurations provide the parameters which calibrate the Hamiltonian. A statistical mechanical analysis shows how the underlying potential properties can be extracted from the distributions found in quenches from stochastic configurations.

We propose a means for constructing highly accurate equations of state (EOS) for elemental solids and liquids essentially from first principles, based upon a particular decomposition of the underlying condensed matter Hamiltonian for the nuclei and electrons. We also point out that at low pressures the neglect of anharmonic and electronphonon terms, both contained in this formalism, results in errors of less than 5% in the thermal parts of the thermodynamic functions. Then we explicitly display the forms of the remaining terms in the EOS, commenting on the use of experiment and electronic structure theory to evaluate them. We also construct an EOS for Aluminum and compare the resulting Hugoniot with data up to 5 Mbar, both to illustrate our method and to see whether the approximation of neglecting anharmonicity et al. remains viable to such high pressures. We find a level of agreement with experiment that is consistent with the low-pressure results.

The proof of the Heisenberg uncertainty relation is modified to produce two improvements: (a) the resulting inequality is stronger because it includes the covariance between the two observables, and (b) the proof lifts certain restrictions on the state to which the relation is applied, increasing its generality. The restrictions necessary for the standard inequality to apply are not widely known, and they are discussed in detail. The classical analog of the Heisenberg relation is also derived, and the two are compared. Finally, the modified relation is used to address the apparent paradox that eigenfunctions of the z component of angular momentum L_z do not satisfy the \phi-L_z Heisenberg relation; the resolution is that the restrictions mentioned above make the usual inequality inapplicable to these states. The modified relation does apply, however, and it is shown to be consistent with explicit calculations.Comment: 12 pages, no figures. Contains corrections to errors in the published editio