Wm. G. Hoover

Two‐dimensional computer studies of crystal stability and fluid viscosity

The mathematical analogy between the elastic stress due to particle displacements in Hookes law solids and the viscous stress due to velocity gradients in incompressible fluids correlates two interesting phenomena. In a two‐dimensional crystal the elastic restoring force opposing particle displacements approaches zero with increasing crystal size, leading to a logarithmically diverging rms displacement in the large‐system limit. The vanishing of the solid‐phase force is mathematically analogous to the lack of viscous damping for a particle moving slowly through a two‐dimensional incompressible fluid. These two continuum results are compared with discrete‐particle computer simulations of two‐dimensional solids and fluids. The divergence predicted by macroscopic elasticity theory agrees quantitatively with computer results for two‐dimensional harmonic crystals. These same results can also be correlated with Whites experimental study of the viscous resistance to a cylinder (a falling wire) moving slowly th...

Liouville's theorems, Gibbs' entropy, and multifractal distributions for nonequilibrium steady states

Liouville’s best-known theorem, ḟ({q,p},t)=0, describes the incompressible flow of phase-space probability density, f({q,p},t). This incompressible-flow theorem follows directly from Hamilton’s equations of motion. It applies to simulations of isolated systems composed of interacting particles, whether or not the particles are confined by a box potential. Provided that the particle–particle and particle–box collisions are sufficiently mixing, the long-time-averaged value 〈f〉 approaches, in a “coarse-grained” sense, Gibbs’ equilibrium microcanonical probability density, feq, from which all equilibrium properties follow, according to Gibbs’ statistical mechanics. All these ideas can be extended to many-body simulations of deterministic open systems with nonequilibrium boundary conditions incorporating heat transfer. Then Liouville’s compressible phase-space-flow theorem—in the original ḟ≠0 form—applies. I illustrate and contrast Liouville’s two theorems for two simple nonequilibrium systems, in each case co...

Isomorphism linking smooth particles and embedded atoms

Macroscopic continuum simulations can be based on an unstructured moving spatial grid made up of “smooth particles”. The smooth particles’ equations of motion include interpolated values of the macroscopic stress gradient at each particle’s position. Microscopic solid-state simulations can be based on the motion of “embedded atoms”, with equations of motion based on a physical idea – embedding atoms in the local electronic density. The embedded atoms then move according to Newtonian equations of motion, based on electronic density gradients at each particle position. I show here that these two descriptions, macroscopic smooth particles and microscopic embedded atoms, can give identical particle trajectories. This demonstration facilitates the understanding of macroscopic models for surface tension and also suggests that certain macroscopic continuum approaches to smooth particle applied mechanics could have useful analogs in microscopic molecular dynamics.

Links between microscopic and macroscopic fluid mechanics

The microscopic and macroscopic versions of fluid mechanics differ qualitatively. Microscopic particles obey time-reversible ordinary differential equations. The resulting particle trajectories {q(t)} may be time-averaged or ensemble-averaged so as to generate field quantities corresponding to macroscopic variables. On the other hand, the macroscopic continuum fields described by fluid mechanics follow irreversible partial differential equations. Smooth particle methods bridge the gap separating these two views of fluids by solving the macroscopic field equations with particle dynamics that resemble molecular dynamics. Recently, nonlinear dynamics have provided some useful tools for understanding the relationship between the microscopic and macroscopic points of view. Chaos and fractals play key roles in this new understanding. Non-equilibrium phase-space averages look very different from their equilibrium counterparts. Away from equilibrium the smooth phase-space distributions are replaced by fractional-dimensional singular distributions that exhibit time irreversibility.

“What is ‘liquid’? Understanding the states of matter”

Molecular dynamics answers the question “what is ‘liquid’?” by describing the detailed dynamic structure of simulated liquids in the many-dimensional phase space of statistical mechanics. The Lyapunov instabilities of liquid motion reveal collective dynamic modes, quite unlike those of solid state studies, superimposed on the van der Waals hard particle static structure. Here we illustrate these developments in the characterization of ‘liquid’ with examples from our recent dynamic instability studies of two-dimensional dumbbell fluids.

Lyapunov Modes of Two-Dimensional Many-Body Systems; Soft Disks, Hard Disks, and Rotors

The dynamical instability of many-body systems can best be characterized through the local Lyapunov spectrum {λ}, its associated eigenvectors {δ}, and the time-averaged spectrum {〈λ〉}. Each local Lyapunov exponent λ describes the degree of instability associated with a well-defined direction—given by the associated unit vector δ—in the full many-body phase space. For a variety of hard-particle systems it is by now well-established that several of the δ vectors, all with relatively-small values of the time-averaged exponent 〈λ〉, correspond to quite well-defined long-wavelength “modes.” We investigate soft particles from the same viewpoint here, and find no convincing evidence for corresponding modes. The situation is similar—no firm evidence for modes—in a simple two-dimensional lattice-rotor model. We believe that these differences are related to the form of the time-averaged Lyapunov spectrum near 〈λ〉=0.

Chaos and irreversibility in simple model systems

The multifractal link between chaotic time-reversible mechanics and thermodynamic irreversibility is illustrated for three simple chaotic model systems: the Baker Map, the Galton Board, and many-body color conductivity. By scaling time, or the momenta, or the driving forces, it can be shown that the dissipative nature of the three thermostated model systems has analogs in conservative Hamiltonian and Lagrangian mechanics. Links between the microscopic nonequilibrium Lyapunov spectra and macroscopic thermodynamic dissipation are also pointed out. (c) 1998 American Institute of Physics.

Nonequilibrium molecular dynamics: the first 25 years

Equilibrium molecular dynamics has been generalized to simulate nonequilibrium systems by adding sources of thermodynamic heat and work. This generalization incorporates microscopic mechanical definitions of macroscopic thermodynamic and hydrodynamic variables, such as temperature and stress, and augments atomistic forces with special boundary, constraint and driving forces capable of doing work on, and exchanging heat with, an otherwise Newtonian system: p ≡ FA(q) + FB(q) + FC(q, p) + FD(q,p) ≡ m(qt+dt −2qt + qt−dt)dt2. The underlying Lyapunov instability of these nonequilibrium equations of motion links microscopic time-reversible deterministic trajectories to macroscopic time-irreversible hydrodynamic behavior as described by the second law of thermodynamics.

Equilibrium and nonequilibrium thermomechanics for an effective pair potential used in smooth particle applied mechanics

The smooth-particle weighting functions used in numerical solutions of the thermomechanical continuum equations can be interpreted as weak pair potentials from the standpoint of statistical physics. We examine both equilibrium and nonequilibrium thermomechanical properties of many-body systems using a typical smooth particle potential, Lucys, and discuss the implications for macroscopic continuum simulations.

Single-Speed Molecular Dynamics of Hard Parallel Squares and Cubes

The fluid and solid equations of state for hard parallel squares and cubes are reinvestigated here over a wide range of densities. We use a novel single-speed version of molecular dynamics. Our results are compared with those from earlier simulations, as well as with the predictions of the virial series, the cell model, and Kirkwood’s many-body single-occupancy model. The single-occupancy model is applied to give the absolute entropy of the solid phases just as was done earlier for hard disks and hard spheres. As we should expect, the excellent agreement found here with all relevant previous work shows very clearly that configurational properties, such as the equation of state, do not require the maximum-entropy Maxwell-Boltzmann velocity distribution. For both hard squares and hard cubes the free-volume theory provides a good description of the high-density solid-phase pressure. Hard parallel squares appear to exhibit a second-order melting transition at a density of 0.79 relative to close-packing. Hard parallel cubes have a more complicated equation of state, with several relatively-gentle curvature changes, but nothing so abrupt as to indicate a first-order melting transition. Because the number-dependence for the cubes is relatively large the exact nature of the cube transition remains unknown.