Nonequilibrium Time Reversibility with Maps and Walks.

Time-reversible dynamical simulations of nonequilibrium systems exemplify both Loschmidt's and Zermélo's paradoxes. That is, computational time-reversible simulations invariably produce solutions consistent with the Second Law of Thermodynamics (Loschmidt's) as well as in the time (Zermélo's, illustrating Poincaré recurrence). Understanding these paradoxical aspects of time-reversible systems is enhanced here by studying the simplest pair of such model systems.

From hard spheres and cubes to nonequilibrium maps with thirty-some years of thermostatted molecular dynamics.

This is our current research perspective on models providing insight into statistical mechanics. It is necessarily personal, emphasizing our own interest in simulation as it developed from the National Laboratories' work to the worldwide explosion of computation of today. We contrast the past and present in atomistic simulations, emphasizing those simple models that best achieve reproducibility and promote understanding.

Heat conduction, and the lack thereof, in time-reversible dynamical systems: generalized Nosé-Hoover oscillators with a temperature gradient.

We use nonequilibrium molecular dynamics to analyze and illustrate the qualitative differences between the one-thermostat and two-thermostat versions of equilibrium and nonequilibrium (heat-conducting) harmonic oscillators. Conservative nonconducting regions can coexist with dissipative heat conducting regions in phase space with exactly the same imposed temperature field.

Shock-wave compression and Joule-Thomson expansion.

Structurally stable atomistic one-dimensional shock waves have long been simulated by injecting fresh cool particles and extracting old hot particles at opposite ends of a simulation box. The resulting shock profiles demonstrate tensor temperature, Txx≠Tyy and Maxwell's delayed response, with stress lagging strain rate and heat flux lagging temperature gradient. Here this same geometry, supplemented by a short-ranged external "plug" field, is used to simulate steady Joule-Kelvin throttling flow of hot dense fluid through a porous plug, producing a dilute and cooler product fluid.

Time-reversal symmetry and covariant Lyapunov vectors for simple particle models in and out of thermal equilibrium.

Recently, a new algorithm for the computation of covariant Lyapunov vectors and of corresponding local Lyapunov exponents has become available. Here we study the properties of these still unfamiliar quantities for a simple model representing a harmonic oscillator coupled to a thermal gradient with a two-stage thermostat, which leaves the system ergodic and fully time reversible. We explicitly demonstrate how time-reversal invariance affects the perturbation vectors in tangent space and the associated local Lyapunov exponents.

Well-posed two-temperature constitutive equations for stable dense fluid shock waves using molecular dynamics and generalizations of Navier-Stokes-Fourier continuum mechanics.

Guided by molecular dynamics simulations, we generalize the Navier-Stokes-Fourier constitutive equations and the continuum motion equations to include both transverse and longitudinal temperatures. To do so we partition the contributions of the heat transfer, the work done, and the heat flux vector between the longitudinal and transverse temperatures. With shockwave boundary conditions time-dependent solutions of these equations converge to give stationary shockwave profiles.

Tensor temperature and shock-wave stability in a strong two-dimensional shock wave.

The anisotropy of temperature is studied here in a strong two-dimensional shock wave, simulated with conventional molecular dynamics. Several forms of the kinetic temperature are considered, corresponding to different choices for the local instantaneous stream velocity. A local particle-based definition omitting any "self"-contribution to the stream velocity gives the best results.

Nonlinear stresses and temperatures in transient adiabatic and shear flows via nonequilibrium molecular dynamics: Three definitions of temperature.

We compare nonlinear stresses and temperatures for adiabatic-shear flows, using up to 262, 144 particles, with those from corresponding homogeneous and inhomogeneous flows. Two varieties of kinetic temperature tensors are compared to the configurational temperatures. This comparison of temperatures led us to two findings beyond our original goal of analyzing shear algorithms.

Microscopic and macroscopic stress with gravitational and rotational forces.

Many recent papers have questioned Irving and Kirkwood's atomistic expression for stress. In Irving and Kirkwood's approach both interatomic forces and atomic velocities contribute to stress. It is the velocity-dependent part that has been disputed.

Simulation of two- and three-dimensional dense-fluid shear flows via nonequilibrium molecular dynamics: comparison of time-and-space-averaged stresses from homogeneous Doll's and Sllod shear algorithms with those from boundary-driven shear.

Homogeneous shear flows (with constant strainrate dv(x)/dy) are generated with the Doll's and Sllod algorithms and compared to corresponding inhomogeneous boundary-driven flows. We use one-, two-, and three-dimensional smooth-particle weight functions for computing instantaneous spatial averages. The nonlinear normal-stress differences are small, but significant, in both two and three space dimensions.

Nonequilibrium temperature and thermometry in heat-conducting phi4 models.

We analyze temperature and thermometry for simple nonequilibrium heat-conducting models. We also show in detail, for both two- and three-dimensional systems, that the ideal-gas thermometer corresponds to the concept of a local instantaneous mechanical kinetic temperature. For the phi4 models investigated here the mechanical temperature closely approximates the local thermodynamic equilibrium temperature.

Hamiltonian dynamics of thermostated systems: two-temperature heat-conducting phi4 chains.

We consider and compare four Hamiltonian formulations of thermostated mechanics, three of them kinetic, and the other one configurational. Though all four approaches "work" at equilibrium, their application to many-body nonequilibrium simulations can fail to provide a proper flow of heat. All the Hamiltonian formulations considered here are applied to the same prototypical two-temperature "phi4" model of a heat-conducting chain.

Smooth-particle phase stability with generalized density-dependent potentials.

Stable fluid and solid particle phases are essential to the simulation of continuum fluids and solids using smooth particle applied mechanics. We show that density-dependent potentials, such as Phi rho=1/2 Sigma(rho-rho 0)2, along with their corresponding constitutive relations, provide a simple means for characterizing fluids and that special stabilization potentials, with density gradients or curvatures, such as Phi inverted Delta rho=1/2 Sigma(inverted Delta rho)2, not only stabilize crystalline solid phases (or meshes) but also provide a surface tension which is missing in the usual density-dependent-potential approach. We illustrate these ideas for two-dimensional square, triangular, and hexagonal lattices.

Smooth-particle applied mechanics: Conservation of angular momentum with tensile stability and velocity averaging.

Smooth-particle applied mechanics (SPAM) provides several approaches to approximate solutions of the continuum equations for both fluids and solids. Though many of the usual formulations conserve mass, (linear) momentum, and energy, the angular momentum is typically not conserved by SPAM. A second difficulty with the usual formulations is that tensile stress states often exhibit an exponentially fast high-frequency short-wavelength instability, "tensile instability.

Smooth-particle boundary conditions.

We study the relative usefulness of static and dynamic boundary conditions as a function of system dimensionality. In one space dimension, dynamic boundaries, with the temperatures and velocities of external mirror-image boundary particles linked directly to temperatures and velocities of interior particles, perform qualitatively better than the simpler static-mirror-image boundary condition with fixed boundary temperatures and velocities. In one space dimension, the Euler-Maclaurin sum formula shows that heat-flux errors with dynamic temperature boundaries vary as h(-4), where h is the range of the smooth-particle weight function w(r View Article and Find Full Text PDF

Lyapunov instability of two-dimensional fluids: Hard dumbbells.

We generalize Benettin's classical algorithm for the computation of the full Lyapunov spectrum to the case of a two-dimensional fluid composed of linear molecules modeled as hard dumbbells. Each dumbbell, two hard disks of diameter sigma with centers separated by a fixed distance d, may translate and rotate in the plane. We study the mixing between these qualitatively different degrees of freedom and its influence on the full set of Lyapunov exponents.

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.

Fluctuations, convergence times, correlation functions, and power laws from many-body Lyapunov spectra for soft and hard disks and spheres.

The dynamical instability of many-body systems is best characterized through the time-dependent local Lyapunov spectrum [lambda(j)], its associated comoving eigenvectors [delta(j)], and the "global" time-averaged spectrum []. We study the fluctuations of the local spectra as well as the convergence rates and correlation functions associated with the delta vectors as functions of j and system size N. All the number dependences can be described by simple power laws.

Chaos, ergodic convergence, and fractal instability for a thermostated canonical harmonic oscillator.

The authors thermostat a qp harmonic oscillator using the two additional control variables zeta and xi to simulate Gibbs' canonical distribution. In contrast to the motion of purely Hamiltonian systems, the thermostated oscillator motion is completely ergodic, covering the full four-dimensional [q,p,zeta,xi] phase space. The local Lyapunov spectrum (instantaneous growth rates of a comoving corotating phase-space hypersphere) exhibits singularities like those found earlier for Hamiltonian chaos, reinforcing the notion that chaos requires kinetic-as opposed to statistical-study, both at and away from equilibrium.

Finite-precision stationary states at and away from equilibrium.

We study the precision dependence of equilibrium and nonequilibrium phase-space distribution functions for time-reversible dynamical systems simulated with finite, computational precision. The conservative and dissipative cases show different behavior, with substantially reduced period lengths in the dissipative case. The main effect of finite precision is to change the phase-space fraction occupied by the distributions.