William Graham Hoover

University of California, Davis | UCD · Center for Neuroscience

Some paradoxical aspects of the Nos\'e and Nos\'e-Hoover dynamics of 1984 and Dettmann's dynamics of 1996 are elucidated. Phase-space descriptions of thermostated harmonic oscillator dynamics can be simultaneously expanding, incompressible, or contracting, as is described here by a variety of three- and four-dimensional phase-space models. These fi...

Considerable research has led to ergodic isothermal dynamics which can replicate Gibbs' canonical distribution for simple ( small ) dynamical problems. Adding one or two thermostat forces to the Hamiltonian motion equations can give an ergodic isothermal dynamics to a harmonic oscillator, to a quartic oscillator, and even to the "Mexican-Hat" ( dou...

The 2017 Snook Prize has been awarded to Kenichiro Aoki for his exploration of chaos in Hamiltonian $\phi^4$ models. His work addresses symmetries, thermalization, and Lyapunov instabilities in few-particle dynamical systems. A companion paper by Timo Hofmann and Jochen Merker is devoted to the exploration of generalized H\'enon-Heiles models and h...

This book aims to provide an example-based education in numerical methods for atomistic and continuum simulations of systems at and away from equilibrium. The focus is on nonequilibrium systems, stressing the use of tools from dynamical systems theory for their analysis. Lyapunov instability and fractal dimensionality are introduced and algorithms...

We develop a bit-reversible implementation of Milne's Fourth-order Predictor algorithm so as to generate precisely time-reversible simulations of irreversible processes. We apply our algorithm to the collision of two zero-temperature Morse-potential balls, which collide to form a warm liquid oscillating drop. The oscillations are driven by surface...

Dynamical systems with special properties are continually being proposed and studied. Many of these systems are variants of the simple harmonic oscillator with nonlinear damping. This paper characterizes these systems as a hierarchy of increasingly complicated equations with correspondingly interesting behavior, including coexisting attractors, cha...

The nonequilibrium Time-Reversible Baker Map provides simple illustrations of the Fluctuation Theorem, the Central Limit Theorem, and the Biased Random Walk. This is material in preparation for the Book form of Carol's and my 2016 Kharagpur Lectures.

We point out that two of Milne's fourth-order integrators are well-suited to bit-reversible simulations. The fourth-order method improves on the accuracy of Levesque and Verlet's algorithm and simplifies the definition of the velocity $v$ and energy $e = (q^2 + v^2)/2$ . ( We use this one-dimensional oscillator problem as an illustration throughout...

The time-averaged Lyapunov exponents support a mechanistic description of the chaos generated in and by nonlinear dynamical systems. The exponents are ordered from largest to smallest with the largest one describing the exponential growth rate of the (small) distance between two neighboring phase-space trajectories. Two exponents describe the rate...

The 2016 Snook Prize has been awarded to Diego Tapias, Alessandro Bravetti, and David Sanders for their paper -- Ergodicity of One-Dimensional Systems Coupled to the Logistic Thermostat. They introduced a relatively stiff hyperbolic tangent thermostat force and successfully tested its ability to reproduce Gibbs' canonical distribution for the harmo...

We describe the application of adaptive (variable time step) integrators to stiff differential equations encountered in many applications. Linear harmonic oscillators subject to nonlinear thermal constraints can exhibit either stiff or smooth dynamics. Two closely related examples, Nosé's dynamics and Nosé–Hoover dynamics, are both based on Hamilto...

For a harmonic oscillator, Nos\'e's single-thermostat approach to simulating Gibbs' canonical ensemble with dynamics samples only a small fraction of the phase space. Nos\'e's approach has been improved in a series of three steps: [ 1 ] several two-thermostat sets of motion equations have been found which cover the complete phase space in an ergodi...

We two had year-long research leaves in Japan, working together fulltime with several Japanese plus Tony De Groot back in Livermore and Harald Posch in Vienna. We summarize a few of the high spots from that very productive year ( 1989-1990 ), followed by an additional fifteen years' work in Livermore, with extensive travel. Next came our retirement...

Childhood and graduate school at Ann Arbor Michigan prepared Bill for an interesting and rewarding career in physics. Along the way came Carol and many joint discoveries with our many colleagues to whom we both owe this good life. This summary of Bill's early work prior to their marriage and sabbatical in Japan is Part I, prepared for Bill's 80th B...

We revisit the equilibrium one-dimensional $\phi^4$ model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable while at the same time exhibiting positive Lyapunov exponents. We formulate a standard initial condition for the investigation of the microcanonical chaotic number depende...

Gibbs’ thermodynamic entropy is given by the logarithm of the phase volume, which itself responds to heat transfer to and from thermal reservoirs. We compare the thermodynamic dissipation described by (i) phase-volume loss with (ii) heat-transfer entropy production. Their equivalence is documented for computer simulations of the response of an ergo...

"Stiff" differential equations are commonplace in engineering and dynamical systems. To solve them we need flexible integrators that can deal with rapidly-changing righthand sides. This tutorial describes the application of "adaptive" [ variable timestep ] integrators to "stiff" mechanical problems encountered in modern applications of Gibbs' 1902...

Berni Julian Alder profoundly influenced my research career at the Livermore Laboratory and the Davis Campus' Teller Tech, beginning in 1962 and lasting for over fifty years. I very much appreciate the opportunity provided by his Ninetieth Birthday Celebration to review some of the many high spots along the way.

Nos\'e's pioneering 1984 work inspired a variety of time-reversible deterministic thermostats. Though several groups have developed successful doubly-thermostated models, single-thermostat models have failed to generate Gibbs' canonical distribution for the one-dimensional harmonic oscillator. A 2001 doubly-thermostated model, claimed to be ergodic...

We relate progress in statistical mechanics, both at and far from equilibrium, to advances in the theory of dynamical systems. We consider computer simulations of time-reversible deterministic chaos in small systems with three- and four-dimensional phase spaces. These models provide us with a basis for understanding equilibration and thermodynamic...

The time reversibility characteristic of Hamiltonian mechanics has long been extended to nonHamiltonian dynamical systems modeling nonequilibrium steady states with feedback-based thermostats and ergostats. Typical solutions are multifractal attractor-repellor phase-space pairs with reversed momenta and unchanged coordinates, $(q,p)\longleftrightar...

Although Nos\'e's thermostated mechanics is formally consistent with Gibbs' canonical ensemble, the thermostated Nos\'e-Hoover ( harmonic ) oscillator, with its mean kinetic temperature controlled, is far from ergodic. Much of its phase space is occupied by regular conservative tori. Oscillator ergodicity has previously been achieved by controlling...

Symplectic methods, which are precisely compatible with Liouville's phase-volume-conservation theorem, are often recommended for computational simulations of Hamiltonian mechanics. Lack of energy drift is an advantage of symplectic methods. But all numerical methods are susceptible to chaos, Lyapunov instability, which severely limits the maximum t...

Nos\'e and Hoover's 1984 work showed that although Nos\'e and Nos\'e-Hoover dynamics were both consistent with Gibbs' canonical distribution neither dynamics, when applied to the harmonic oscillator, provided Gibbs' Gaussian distribution. Further investigations indicated that two independent thermostat variables are necessary, and often sufficient,...

The relative stability and ergodicity of deterministic time-reversible thermostats, both singly and in coupled pairs, are assessed through their Lyapunov spectra. Five types of thermostat are coupled to one another through a single Hooke's-Law harmonic spring. The resulting dynamics shows that three specific thermostat types, Hoover-Holian, Ju-Bulg...

Shuichi Nos\'e opened up a new world of atomistic simulation in 1984. He formulated a Hamiltonian tailored to generate Gibbs' canonical distribution dynamically. This clever idea bridged the gap between microcanonical molecular dynamics and canonical statistical mechanics. Until then the canonical distribution was explored with Monte Carlo sampling...

Typical Hamiltonian liquids display exponential "Lyapunov instability", also called "sensitive dependence on initial conditions". Although Hamilton's equations are thoroughly time-reversible the forward and backward Lyapunov instabilities can differ, qualitatively. In numerical work the expected forward/backward pairing of Lyapunov exponents is als...

Structurally-stable atomistic one-dimensional shockwaves 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, with the longitudinal temperature exceeding the transverse, and Maxwell's delayed response, with stres...

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 i...

We consider the local (instantaneous) Lyapunov spectrum for a four-dimensional Hamiltonian system. Its stable periodic motion can be reversed for long times. Its unstable chaotic motion, with two symmetric pairs of exponents, cannot. In the latter case reversal occurs for more than a thousand fourth-order Runge–Kutta time steps, followed by a trans...

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. In some cases conservative and dissipative regions in phase space coexist for exactly the same imposed temperature field.

We provide a simple example of a chaotic thermostated harmonic-oscillator system which exhibits qualitatively different local Lyapunov exponents for simple scale-model constant-volume transformations of its coordinate q and momentum p : {q,p} ----> {(Q/s),(sP)} . The time-dependent thermostat variable zeta(t) is unchanged by such scaling. The origi...

Nearly all the evolution equations of physics are time-reversible, in the sense that a movie of the solution, played backwards, would obey exactly the same differential equations as the original forward solution. By way of contrast, stochastic approaches are typically not time-reversible, though they could be made so by the simple expedient of stor...

We apply Maxwell and Cattaneo's relaxation approaches to the analysis of strong shockwaves in a two-dimensional viscous heat-conducting fluid. Good agreement results for reasonable values of Maxwell's relaxation times. Instability results if the viscous relaxation time is too large. These relaxation terms have negligible effects on slower-paced sub...

Hamiltonian mechanics can be used to constrain temperature simultaneously with energy. We illustrate the interesting situations that develop when two different temperatures are imposed within a composite Hamiltonian system. The model systems we treat are "phi-4" chains, with quartic tethers and quadratic nearest-neighbor Hooke's-law interactions. T...

Hamiltonian trajectories are strictly time-reversible. Any time series of Hamiltonian coordinates {q} satisfying Hamilton's motion equations will likewise satisfy them when played "backwards", with the corresponding momenta changing signs : {+p} --> {-p}. Here we adopt Levesque and Verlet's precisely bit-reversible motion algorithm to ensure that t...

A Comment on the Letter by M. Campisi et al. Phys. Rev. Lett. 108, 250601 (2012). The authors of the Letter offer a Reply.

We discuss the irreversibility, nonlocality, and fluctuations, as well as the Lyapunov and hydrodynamic instabilities characterizing atomistic, smooth-particle, and finite-difference solutions of the two-dimensional Rayleigh-B\'enard problem. To speed up the numerical analysis we control the time-dependence of the Rayleigh number so as to include m...

Molecular Dynamics and Statistical Mechanics make possible a particle-based understanding of Thermodynamics and Hydrodynamics, including the fascinating Loschmidt contradiction between time-reversible atomistic mechanics and the time-irreversible thermodynamic dissipation incorporated into macroscopic fluid and solid mechanics.

Campisi, Zhan, Talkner, and Haenggi state, in promoting a new logarithmic computational thermostat [ arXiv 1203.5968 and 1204.4412 ], that (thermostated) Nose-Hoover mechanics is not Hamiltonian. First I point out that Dettmann clearly showed the Hamiltonian nature of Nose-Hoover mechanics. The trajectories {q(t)} generated by Dettmann's Hamiltonia...

Dufty, Lee, Lutsko, Montanero, and Santos have carried out stability analyses of steady stationary shear flows. Their approach is based on the compressible and heat conducting Navier-Stokes-Fourier model. It predicts the unstable exponential growth of long-wavelength transverse perturbations for both two- and three-dimensional fluids. We point out...

Strong shockwaves generate entropy quickly and locally. The Newton-Hamilton equations of motion, which underly the dynamics, are perfectly time-reversible. How do they generate the irreversible shock entropy? What are the symptoms of this irreversibility? We investigate these questions using Levesque and Verlet's bit-reversible algorithm. In this w...

We compare the Gram-Schmidt and covariant phase-space-basis-vector descriptions for three time-reversible harmonic oscillator problems, in two, three, and four phase-space dimensions respectively. The two-dimensional problem can be solved analytically. The three-dimensional and four-dimensional problems studied here are simultaneously chaotic, time...

We illustrate some of the static and dynamic relations discovered by Cohen, Crooks, Evans, Jarzynski, Kirkwood, Morriss, Searles, and Zwanzig. These relations link nonequilibrium processes to equilibrium isothermal free energy changes and to dynamical path probabilities. We include ideas suggested by Dellago, Geissler, Oberhofer, and Schoell-Paschi...

We apply Maxwell and Cattaneo's relaxation approaches to the analysis of strong shockwaves in a two-dimensional viscous heat-conducting fluid. Good agreement results for reasonable values of Maxwell's relaxation times. Instability results if the viscous relaxation time is too large. These relaxation terms have negligible effects on slower-paced sub...

By borrowing ideas from control theory, Nonequilibrium Molecular Dynamics incorporates temperature, stress, and heat flux directly into atomistic, time-reversible, deterministic equations of motion. We are applying this technique to studies of surface indentation, surface cutting, friction, ablation, and condensation. Here we describe simulations o...

Predicting nonequilibrium fluctuations requires a knowledge of nonequilibrium distribution functions. Despite the distributions' fractal character some theoretical results, "Fluctuation Theorems", reminiscent of but distinct from, Gibbs' equilibrium statistical mechanics and the Central Limit Theorem, have been established away from equilibrium and...

In this work we analyze a recent phenomenological hydrodynamic theory proposed recently by Hoover et al. [arXiv:1005.1525v1] to study shock waves in dense fluids. The theory incorporates anisotropic temperature and relaxation for the fluxes and temperature, following the ideas by Maxwell, Cattaneo, and Krook. For the steady case we analyze the poin...

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 erg...

We discuss three related subjects well suited to graduate research. The first, Nonequilibrium molecular dynamics or "NEMD", makes possible the simulation of atomistic systems driven by external fields, subject to dynamic constraints, and thermostated so as to yield stationary nonequilibrium states. The second subject, Smooth Particle Applied Mechan...

Macroscopic models which distinguish the longitudinal and transverse temperatures can provide improved descriptions of the microscopic shock structures as revealed by molecular dynamics simulations. Additionally, we can include three relaxation times in the models, two based on Maxwell's viscoelasticity and its Cattaneo-equation analog for heat flo...

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 number of simple models, including an harmonic oscillator coupled to a thermal gradient with a two-stage thermostat, which leaves the...

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 temperatur...

Shockwaves provide a useful and rewarding route to the nonequilibrium properties of simple fluids far from equilibrium. For simplicity, we study a strong shockwave in a dense two-dimensional fluid. Here, our study of nonlinear transport properties makes plain the connection between the observed local hydrodynamic variables (like the various gradien...

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 strea...

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 sin...

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...

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. To help clarify this situation we investigate (i) a fluid in a gravitational field and (ii) a steadi...

In the half century since the 1950s computer simulation has transformed our understanding of physics. The rare, expensive, slow, and bulky mainframes of World War II have given way to today's millions of cheap, fast, desksized workstations and personal computers. As a result of these changes, the theoretical formal view of physics has gradually shi...

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, b...

We investigate and discuss the time-reversible nature of phase-space instabilities for several flows, . The flows describe thermostated oscillator systems in from two through eight phase-space dimensions. We determine the local extremal phase-space growth rates, which bound the instantaneous comoving Lyapunov exponents. The extremal rates are point...

Microscopic and macroscopic particle simulation techniques are useful introductions to computational physics. These techniques make it possible to simulate complex problems in fluid and solid mechanics, including laminar and turbulent flows, shockwaves, as well as fracture and failure in solids. We illustrate several particle-based techniques with...

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 appr...

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...

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equili...

The dissipation associated with nonequilibrium flow processes is reflected by the formation of strange attractor distributions in phase space. The information dimension of these attractors is less than that of the equilibrium phase space, corresponding to the extreme rarity of nonequilibrium states. Here we take advantage of a simple model for heat...

Molecular dynamics is limited to small-size short-time situations in which the interparticle forces are known. Continuum mechanics applies at the time and distance scales relevant to men. Smooth particle applied mechanics (SPAM) is a simple, transparent, and flexible approach to solving continuum mechanics problems with particles. Here we point out...

Microscopic and macroscopic particle-simulation methods can both be applied to interesting nonequilibrium problems. Here I develop and discuss the ordinary differential equations underlying these two approaches and illustrate them with applications of interest to statistical mechanics and computational fluid mechanics.

At equilibrium Nosé's 1984 revolutionary thermostat idea linked Newton's mechanics with Gibbs' statistical mechanics. His work expanded the scope of isothermal and isobaric simulations. Nosé-Hoover dynamics has subsequently facilitated the simulation and detailed understanding of nonequilibrium problems. The fractal phase-space distributions, and t...

The same methods are now being applied to solid-phase problems. 10 At the relatively high frequencies used in the viscous fluid calculations described here, solids typically behave elastically. Lower frequencies lead to the formation of dislocations and other defects, making it possible to study plastic flow. A property of the nonequilibrium equati...

In this paper we are concerned with the instability of the phase-space trajectory for particle models resembling classical fluids with short-range interactions. Recently, the application of dynamical systems theory - and the computation of Lyapunov spectra in particular - has provided new and in some respect complementary insight both for systems i...

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 stab...

Atomistic Molecular Dynamics and Lagrangian Continuum Mechanics can be very similarly adapted to massively-parallel computers. Millions of degrees of freedom can be treated. The two complementary approaches, microscopic and macroscopic, are being applied to increasingly realistic flows of fluids and solids. The two approaches can also be combined i...

Nonequilibrium Molecular Dynamics requires an extension of Newtonian and Hamiltonian mechanics. This new extended mechanics includes Gauss' and Nosé's thermostatted equations of motion. Here I review the past 20 years' history of the various formulations, solutions, interpretations, and further extensions of these ``new'' motion equations. I emphas...

The present special issue of physica status solidi (b), guest-edited by Krzysztof W. Wojciechowski, Andrew Alderson, Arkadiusz Brańka, and Kim L. Alderson, is dedicated to Auxetics and Related Systems – materials which exhibit negative Poisson's ratio behaviour. Most papers were presented at a workshop which was held in Poznań–Będlewo, 27–30 June 2...

In my own research career I have been primarily interested in the statistical mechanics of nonequilibrium systems of particles, stressing new techniques for undertaking and understanding computer simulations [1, 2]. The blind alleys toward which molecular dynamics naturally leads provide a compensating appreciation of continuum mechanics, with its...

Nonequilibrium Molecular Dynamics is a powerful simulation tool. Like its equilibrium cousin, nonequilibrium molecular dynamics is based on time-reversible equations of motion. But unlike conventional mechanics, nonequilibrium molecular dynamics provides a consistent microscopic basis for the irreversible macroscopic Second Law of Thermodynamics. W...

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 tens...

Thermostated tethered harmonic lattices provide good illustrations of the phase-space dimensionality loss ΔD which occurs in the strange attractor distributions characterizing stationary nonequilibrium flows. We use time-reversible nonequilibrium molecular dynamics, with two Nosé–Hoover thermostats, one hot and one cold, to study a family of square...

Seven different time-reversible deterministic thermostats are considered here and applied to a simple particle-based nonequilibrium heat-flow problem. This approach is robust. Results for all these different thermostats agree rather well for system widths of ten particle diameters or more. The simplest of the thermostats is the Gauss–Nosé–Hoover th...

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...

MOLECULAR DYNAMICS has been generalized in order to stimulate a variety of NONEQUILIBRIUM systems. This generalization has been achieved by adopting microscopic mechanical definitions of macroscopic thermodynamic and hydrodynamic variables, such as temperature and stress. Some of the problems already treated include rapid plastic deformation, inten...

The use of molecular dynamics to explain mechanisms of flows is examined. The present theory on flows far from equilibrium is described. Various types of atomistic computer simulation techniques and their application to modeling of a heat flow in a three-body one-dimensional periodic chain are investigated. Methods of controlling hydrodynamic and t...

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 si...

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 macroscop...

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 [<lambda(j)>]. We study the fluctuations of the local spectra as well as the convergence rates and correlation functions associa...

Describes the work of physicists, chemists, mathematicians, and engineers who are attacking the classic problem of the reversibility paradox, describing their work from the perspective of computer simulation, emphasizing the author's approach to the problem of understanding the compatability of time-reverse mechanics.

In this paper, we implement a SPAM (Smooth Particle Applied Mechanics) code in both pure MPI and MPI-OpenMP hybrid manner, then compare and analize the performance of them on an SMPPC cluster. Our SPAM code is described to handle any mapping of spatial cells on to parallel MPI processes to exploit well load-balancing even with a relatively high com...

The dynamical instability of many-body systems can best be characterized through the local Lyapunov spectrum {}, its associated eigenvectors {}, and the time-averaged spectrum {}. Each local Lyapunov exponent describes the degree of instability associated with a well-defined direction—given by the associated unit vector —in the full many-body phase...

Forward and backward trajectories from time-symmetric equations of motion can have time-asymmetric stability properties, and exhibit time-asymmetric fluctuations. Away from equilibrium this symmetry breaking is the mechanical equivalent of the second law of thermodynamics. Strange attractor states obeying the second law are time-reversed versions o...

Smooth Particle Applied Mechanics (SPAM) is a technique which provides a versatile approach to simulating many difficult problems in continuum mechanics, such as the breakup of a cavitating fluid and the penetration of one solid by another. SPAM also provides a simple evaluation method for all the continuum variables, as well as the spatial gradien...

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...