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Aspects of the Nos\'e and Nos\'e-Hoover dynamics developed in 1983-1984 along with Dettmann's closely related dynamics of 1996, are considered. We emphasize paradoxes associated with Liouville's Theorem. Our account is pedagogical, focused on the harmonic oscillator for simplicity, though exactly the same ideas can be, and have been, applied to manybody systems. Nos\'e, Nos\'e-Hoover, and Dettmann flows were all developed in order to access Gibbs' canonical ensemble directly from molecular dynamics. Unlike Monte Carlo algorithms dynamical flow models are often not ergodic and so can fail to reproduce Gibbs' ensembles. Accordingly we include a discussion of ergodicity, the visiting of all relevant microstates corresponding to the desired ensemble. We consider Lyapunov instability too, the usual mechanism for phase-space mixing. We show that thermostated harmonic oscillator dynamics can be simultaneously expanding, incompressible, or contracting, depending upon the chosen "phase space". The fractal nature of nonequilibrium flows is also illustrated for two simple two-dimensional models, the hard-disk-based Galton Board and the time-reversible Baker Map. The simultaneous treatment of flows as one-dimensional and many-dimensional suggests some interesting topological problems for future investigations., Comment: 41 pages with 11 figures, and with minor typographical errors and Baker-Map results corrected, accepted for publication in Computational Methods in Science and Technology (20 August, 2019) with proof corrections of 19 September

The one-dimensional $\phi^4$ Model generalizes a harmonic chain with nearest-neighbor Hooke's-Law interactions by adding quartic potentials tethering each particle to its lattice site. In their studies of this model Kenichiro Aoki and Dimitri Kusnezov emphasized its most interesting feature : because the quartic tethers act to scatter long-wavelength phonons, $\phi^4$ chains exhibit Fourier heat conduction. In his recent Snook-Prize work Aoki also showed that the model can exhibit chaos on the three-dimensional energy surface describing the two-body two-spring chain. That surface can include {\it at least two} distinct chaotic seas. Aoki pointed out that the model typically exhibits different kinetic temperatures for the two bodies. Evidently few-body $\phi^4$ problems merit more investigation. Accordingly, the 2018 Prizes honoring Ian Snook (1945-2013) will be awarded to the author(s) of the most interesting work analyzing and discussing few-body $\phi^4$ models from the standpoints of dynamical systems theory and macroscopic thermodynamics, taking into account the model's ability to maintain a steady-state kinetic temperature gradient as well as at least two coexisting chaotic seas in the presence of deterministic chaos., Comment: Eight pages with five figures prepared for Computational Methods in Science and Technology, Volume 24(2)

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 has been selected for Honorable Mention in the Snook-Prize competition., Comment: Ten pages with four figures

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 stress lagging strainrate 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., Comment: Eight pages and three figures. Comments welcome

We explore 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 original (q,p,zeta) motion and the scale-model (Q,P,zeta) version of the motion are physically identical. But both the local Gram-Schmidt Lyapunov exponents and the related local "covariant" exponents change with the change of scale. Thus this model furnishes a clearcut chaotic time-reversible example showing how and why both the local Lyapunov exponents and covariant exponents vary with the scale factor s ., Comment: Eight pages with three figures, as submitted to Computational Methods in Science and Technology

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 storing their underlying pseudorandom numbers in an array. Here we illustrate the notion of time-reversible random number generators. In Version 1 we offered a suitable reward for the first arXiv response furnishing a reversed version of an only slightly-more-complicated pseudorandom number generator. Here we include Professor Ricci-Tersenghi's prize-winning reversed version as described in his arXiv:1305.1805 contribution: "The Solution to the Challenge in `Time-Reversible Random Number Generators' by Wm. G. Hoover and Carol G. Hoover"., Comment: Seven pages with a single Figure, dedicated to the memories of our late colleague Ian Snook

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. This model is known to satisfy Fourier's law. Our prototypical problem sandwiches a Newtonian subsystem between hot and cold Hamiltonian reservoir regions. We have characterized four different Hamiltonian reservoir types. There is no tendency for any of these two-temperature Hamiltonian simulations to transfer heat from the hot to the cold degrees of freedom. Evidently steady heat flow simulations require energy sources and sinks, and are therefore incompatible with Hamiltonian mechanics., Comment: 17 pages with four figures ; see also the many arXiv contributions mentioned in the references. Published in Communications in Nonlinear Science and Numerical Simulation 18, 3365-3372(2013); (received and accepted 10 May 2013, available online 17 May 2013)

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 the trajectory reversibility is exact, with the forward and backward sets of coordinates identical. Nevertheless, the associated instantaneous Lyapunov instability, or "sensitive dependence on initial conditions" of "chaotic" (or "Lyapunov unstable") bit-reversible coordinate trajectories can still exhibit an exponentially growing time-symmetry-breaking irreversibility. Surprisingly, the positive and negative exponents, as well as the forward and backward Lyapunov spectra, are usually not closely related, and so give four differing topological measures of "local" chaos. We have demonstrated this symmetry breaking for fluid shockwaves, for free expansions, and for chaotic molecular collisions. Here we illustrate and discuss this time-symmetry breaking for three statistical-mechanical systems, [1] a minimal (but still chaotic) one-body "cell model" with a four-dimensional phase space; [2] relatively small colliding crystallites, for which the whole Lyapunov spectrum is accessible; [3] a near-continuum inelastic collision of two larger 400-particle balls. In the last two of these pedagogical problems the two colliding bodies coalesce. The particles most prone to Lyapunov instability are dramatically different in the two time directions. Thus this Lyapunov-based symmetry breaking furnishes an interesting Arrow of Time., Comment: 23 pages with 13 figures

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 many distinct flow morphologies in a single simulation. The relatively simple nature of these computational algorithms and the richness of the results they can yield make such studies and their interpretation particularly well suited to graduate-level research., Comment: Sixteen Pages and Eight Figures -- for Presentation at Saint Petersburg Russia at the 40th Advanced Problems in Mechanics Conference, July 2012

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., Comment: Eighteen pages with six figures. Contribution to the Advanced Problems in Mechanics 40th Conference in Saint Petersburg (Russia). Two typos fixed

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 that the patently-stable two-dimensional periodic shear flows studied earlier by Petravic, Posch, and ourselves contradict these predicted instabilities. The stable steady-state shear flows are based on nonequilibrium molecular dynamics with simple thermostats maintaining nonequilibrium stationary states in two space dimensions. The failure of the stability analyses remains unexplained., Comment: Fourteen pages, four figures. For discussion at the Advanced Problems in Mechanics Summer School XXXX at Saint Petersburg, Summer 2012; 1983 Evans-Morriss reference added

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 way we can generate an entirely imaginary past consistent with the irreversibility observed in the present. We use Runge-Kutta integration to analyze the local Lyapunov instability of the forward and backward processes so as to identify those particles most intimately connected with the irreversibility described by the Second Law of Thermodynamics. Despite the perfect time symmetry of the particle trajectories, the fully-converged vectors associated with the largest Lyapunov exponents, forward and backward in time, are qualitatively different. The vectors display a time-symmetry breaking equivalent to Time's Arrow. That is, in autonomous Hamiltonian shockwaves the largest local Lyapunov exponents, forward and backward in time, are quite different., Comment: 14 pages, 4 figures, with responses for the three referees, originally written for a gestating special issue of Journal of Physics A [ under "review" there for the past 13 months, withdrawn today and submitted to Computational Methods in Science and Technology 29 January 2013. ]

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-reversible, and dissipative. Our treatment is intended to be pedagogical, for publication in Communications in Nonlinear Science and Numerical Computation and for use in an updated version of our book on Time Reversibility, Computer Simulation, and Chaos. Comments are very welcome., Comment: 25 pages with 12 figures; New Figures 9-12 based on two billion timesteps rather than the two hundred million used in Version 1; Electronic publication in Communications in Nonlinear Science and Numerical Computation scheduled for 1 July 2011

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-Paschinger. Our treatment is intended to be pedagogical, for use in an updated version of our book: Time Reversibility, Computer Simulation, and Chaos. Comments are very welcome., Comment: 11 pages

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 subsonic problems, as is shown here for two-roll and four-roll Rayleigh-Benard flows., Comment: 21 Pages, with six figures, submitted to Physics Letters A

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 applied to simple models. We summarize the simplest of these results for a Gaussian-thermostated Galton Board problem, a field-driven mass point moving through a periodic array of hard-disk scatterers. The billion-collision trillion-timestep data we analyze correspond to periodic orbits with up to 793,951,594 collisions and 447,064,397,614 timesteps., Comment: 15 pages, 6 figures, prepared for second edition of Time Reversibility, Computer Simulation, and Chaos. Comments and suggestions welcome

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 Mechanics or "SPAM", provides a particle method, resembling molecular dynamics, but designed to solve continuum problems. The numerical work is simplified because the SPAM particles obey ordinary, rather than partial, differential equations. The interpolation method used with SPAM is a powerful interpretive tool converting point particle variables to twice-differentiable field variables. This interpolation method is vital to the study and understanding of the third research topic we discuss, strong shockwaves in dense fluids. Such shockwaves exhibit stationary far-from-equilibrium states obtained with purely reversible Hamiltonian mechanics. The SPAM interpolation method, applied to this molecular dynamics problem, clearly demonstrates both the tensor character of kinetic temperature and the time-delayed response of stress and heat flux to the strain rate and temperature gradients. The dynamic Lyapunov instability of the shockwave problem can be analyzed in a variety of ways, both with and without symmetry in time. These three subjects suggest many topics suitable for graduate research in nonlinear nonequilibrium problems., Comment: 40 pages, with 21 figures, as presented at the Granada Seminar on the Foundations of Nonequilibrium Statistical Physics, 13-17 September, as three lectures

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 flow, and a third thermal, based on the Krook-Boltzmann equation. This approach can replicate the observed lags of stress (which lags behind the strain rate) and heat flux (which lags behind the temperature gradient), as well as the eventual equilibration of the two temperatures. For profile stability the time lags cannot be too large. By partitioning the longitudinal and transverse contributions of work and heat and including a tensor heat conductivity and bulk viscosity, all the qualitative microscopic features of strong simple-fluid shockwave structures can be reproduced., Comment: 19 pages with 8 figures, for the Proceedings of Advanced Problems in Mechanics--1-5 July 2010, sponsored by the Institute for Problems in Mechanical Engineering under the patronage of the Russian Academy of Sciences

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. The profiles include anisotropic temperature and can be fitted to molecular dynamics results, demonstrating the utility and simplicity of a two-temperature description of far-from-equilibrium states., Comment: 19 pages with 10 figures, revised following review at Physical Review E and with additional figure/discussion, for presentation at the International Summer School and Conference "Advanced Problems in Mechanics" [Saint Petersburg, Russia] 1-5 July 2010.

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 gradients and fluxes) and the chosen recipes for defining (or "measuring") those variables. The range over which nonlocal hydrodynamic averages are computed turns out to be much more significant than are the other details of the averaging algorithms. The results show clearly the incompatibility of microscopic time-reversible cause-and-effect dynamics with macroscopic instantaneously-irreversible models like the Navier-Stokes equations., Comment: 13 pages with 9 figures, in honor of Leopoldo Garcia-Colin's 80th Birthday. Figures 8 and 9 recalculated using one-dimensional weighting functions in the averages. The momentum flux on page 5 is 9/2, not 5/2

The anisotropy of temperature is studied here in a strong two-dimensional shockwave, 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. The configurational temperature is not useful for this shockwave problem. Configurational temperature is subject to a shear instability and can give local negative temperatures in the vicinity of the shock front. The decay of sinusoidal shockfront perturbations shows that strong two-dimensional planar shockwaves are stable to such perturbations., Comment: 15 pages with 8 figures, after review at Physical Review E

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. 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., Comment: 11 Figures and 28 pages, in preparation for the Journal of Statistical Physics

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 [1] a fluid in a gravitational field and [2] a steadily rotating solid. For both problems we choose conditions where the two stress contributions, potential and kinetic, are significant. The analytic force-balance solutions of both these problems agree very well with a smooth-particle interpretation of the atomistic Irving-Kirkwood stress tensor., Comment: Fifteen pages with seven figures, revised according to referees' suggestions at Physical Review E. See also Liu and Qiu's arXiv contribution 0810.0803

Homogeneous shear flows (with constant strainrate du/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 stress differences are small, but significant, in both two and three space dimensions. In homogeneous systems the sign and magnitude of the shearplane stress difference, P(xx) - P(yy), depend on both the thermostat type and the chosen shearflow algorithm. The Doll's and Sllod algorithms predict opposite signs for this stress difference, with the Sllod approach definitely wrong, but somewhat closer to the (boundary-driven) truth. Neither of the homogeneous shear algorithms predicts the correct ordering of the kinetic temperatures, T(xx) > T(zz) > T(yy)., Comment: 34 pages with 12 figures, under consideration by Physical Review E

We analyze temperature and thermometry for simple nonequilibrium heat-conducting models. We 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 Phi-4 models investigated here the mechanical temperature closely approximates the local thermodynamic equilibrium temperature. There is a significant difference between kinetic temperature and the nonlocal configurational temperature. Neither obeys the predictions of extended irreversible thermodynamics. Overall, we find that kinetic temperature, as modeled and imposed by the Nos\'e-Hoover thermostats developed in 1984, provides the simplest means for simulating, analyzing, and understanding nonequilibrium heat flows., Comment: 20 pages with six figures, revised following review at Physical Review E

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 "phi-4" model of a heat-conducting chain. This model incorporates nearest-neighbor Hooke's-Law interactions plus a quartic tethering potential. Physically correct results, obtained with the isokinetic Gaussian and Nose-Hoover thermostats, are compared with two other Hamiltonian results. The latter results, based on constrained Hamiltonian thermostats, fail correctly to model the flow of heat., Comment: 15 pages, 6 figures, in honor of Prof Harald Posch Birthday #65

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=(1/2)Sum (rho-rho_0)^2, along with their corresponding constitutive relations, provide a simple means for characterizing fluids and that a special stabilization potential, Phi=(1/2)Sum (delrho)^2, not only stabilizes crystalline solid phases (or meshes) but also provides 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., Comment: 10 pages, 5 figures