arXiv subjects

William Graham Hoover

Publications and source records attributed to William Graham Hoover.

At least 19 recordsLinked to original sources

arXiv2025

Chaos in Nonequilibrium Two-Temperature $(T_x, T_y)$ Nosé-Hoover Cell Models

We revisit a two-temperature Nosé-Hoover wanderer particle embedded in a two-dimensional periodic 2x2 cell with four smooth repulsive corners at $(x,y) = (\pm 1, \pm 1)$ to explore chaos with anisotropic thermostatting. The model employs separate thermostats in the x and y directions, enabling controlled deviations from equilibrium. By integrating the full six-dimensional equations of motion and computing the complete Lyapunov spectrum, we confirm chaos and quantify phase-space contraction with high numerical precision. The total contraction rate, interpreted as entropy production, increases nonlinearly with the thermostat anisotropy, deviating from the quadratic dependence expected from linear-response theory, $Λ\propto -δ^{2}$. We compare two fits for $Λ$ as a function of $δ= 0.5 -T_y$: 1) a power law, $Λ\propto -δ^{2.44}$, 2) a quadratic-plus-quartic expansion. While the former captures low-driving behavior slightly better, the latter more accurately describes the strongly driven regime and remains consistent with linear response theory near equilibrium. An empirical linear relation between dissipation and phase-space dimensionality loss is also identified, $Λ\propto (D_{KY}-6) / 3$, where $D_{KY}$ is the approximate Kaplan-Yorke dimension. Our results demonstrate that nonlinear dissipation scaling emerges naturally even in minimal driven systems. Momentum statistics show significant non-Gaussian behavior under strong driving. Despite its dissipative nature, the model remains strictly time-reversible, offering a pedagogically rich example of microscopic reversibility coexisting with macroscopic entropy production.

cond-mat.stat-mech↗

arXiv2024

Canonical Temperature Control by Molecular Dynamics

"Pedagogical derivations for Nosé's dynamics can be developed in two different ways, (i) by starting with a temperature-dependent Hamiltonian in which the variable $s$ scales the time or the mass, or (ii) by requiring that the equations of motion generate the canonical distribution including a Gaussian distribution in the friction coefficient $ζ$. Nosé's papers follow the former approach. Because the latter approach is not only constructive and simple, but also can be generalized to other forms of the equations of motion, we illustrate it here. We begin by considering the probability density $f(q,p,ζ)$ in an extended phase space which includes $ζ$ as well as all pairs of phase variables $q$ and $p$. This density $f(q,p,ζ)$ satisfies the conservation of probability (Liouville's Continuity Equation)" $$(\partial f/\partial t) + \sum (\partial (\dot q f)/\partial q) + \sum (\partial (\dot p f)/\partial p) + \sum (\partial (\dot ζf)/\partial ζ) = 0 \ . $$ The multi-authored ``review''\cite{b1} motivated our quoting the history of Nosé and Nosé-Hoover mechanics, aptly described on page 31 of Bill's 1986 {\it Molecular Dynamics} book, reproduced above\cite{b2}.

cond-mat.stat-mech↗

arXiv2023

Random Walk Equivalence to the Compressible Baker Map and the Kaplan-Yorke Approximation to Its Information Dimension

Simple time-reversible systems can generate {\it irreversible} flows satisfying the Second Law of Thermodynamics. Maps, and equivalent random walks, can also do this. We study a pair of time-reversible Baker Maps, $N2$ and $N3$, which generate dissipative{\it fractal} phase-space structures. Steadily decreasing phase-space volumes correspond to the dissipation associated with entropy production. Like three smooth reversible dissipative one-body phase-space flows developed in the 1980s and 1990s our maps generate fractal distributions, but in two dimensions rather than three, simplifying visualization and analyses. The continuity equation, which quantifies phase-volume loss, motivates study of the fractals' reduced ``information dimensions'', which were approximated by Kaplan and Yorke in terms of two-dimensional maps' two Lyapunov exponents. The maps studied here generate fractal (fractional dimensional) distributions in their phase spaces. By mapping uniformly dense grids of points, fractal dimensions can be determined by ``area-wise'' mappings. Beginning with a uniform grid area-wise mapping of the $N2$ Baker Map provides an information dimension of 1.78969. Alternatively, as many as a trillion iterations, starting from an arbitrary point, gives a smaller ``point-wise'' dimensionality, $1.741_5$. Neither of these precisely determined estimates matches the Kaplan-Yorke conjecture value, 1.7337. In the course of studying these three different approaches to information dimension we developed random walk equivalents to both mappings, which greatly simplifies analyses. We found that for the older $ N2$ Baker map the three approaches all disagree with one another! We later discovered that for the newer $N3$ Baker mapping the three approaches to information dimension, area-wise, point-wise and Kaplan-Yorke, agree.

nlin.CD↗

arXiv2022

Time-Reversible Thermodynamic Irreversibility : One-Dimensional Heat-Conducting Oscillators and Two-Dimensional Newtonian Shockwaves

We analyze the time-reversible mechanics of two irreversible simulation types. The first is a dissipative one-dimensional heat-conducting oscillator exposed to a temperature gradient in a three-dimensional phase space with coordinate $q$, momentum $p$, and thermostat control variable $ζ$. The second type simulates a conservative two-dimensional $N$-body fluid with $4N$ phase variables $\{q,p\}$ undergoing shock compression. Despite the time-reversibility of each of the three oscillator equations and all of the $4N$ manybody motion equations both types of simulation are irreversible, obeying the Second Law of Thermodynamics. But for different reasons. The irreversible oscillator seeks out an attractive dissipative limit cycle. The likewise irreversible, but thoroughly conservative, Newtonian shockwave eventually generates a reversible near-equilibrium pair of rarefaction fans. Both problem types illustrate interesting features of Lyapunov instability.

physics.comp-ph↗

arXiv2022

Thermodynamic Entropy from Sadi Carnot's Cycle using Gauss' and Doll's-Tensor Molecular Dynamics

Carnot's four-part ideal-gas cycle includes both isothermal and adiabatic expansions and compressions. Analyzing this cycle provides the fundamental basis for statistical thermodynamics. We explore the cycle here from a pedagogical view in order to promote understanding of the macroscopic thermodynamic entropy, the state function associated with thermal energy changes. From the alternative microscopic viewpoint the Hamiltonian ${\cal H}(q,p)$ is the energy and entropy is the (logarithm of the) phase-space volume $Ω$ associated with a macroscopic state. We apply two novel forms of Hamiltonian mechanics to Carnot's Cycle: [1] Gauss' isokinetic mechanics for the isothermal segments and [2] Doll's Tensor for the isentropic adiabatic segments. We explore the equivalence of the microscopic and macroscopic views of Carnot's cycle for simple fluids here, beginning with the ideal Knudsen gas and extending the analysis to a prototypical simple fluid.

cond-mat.stat-mech↗

arXiv2022

Computational Implementation of Maxwell's Knudsen-Gas Demon and Its Extension to a Two-Dimensional Soft-Disk Fluid

An interesting preprint by Puru Gujrati, "Maxwell's Demon Must Remain Subservient to Clausius' Statement" [ of the Second Law of Thermodynamics ], traces the development and application of Maxwell's Demon. He argues against the Demon on thermodynamic grounds. Gujrati introduces and uses his own version of a generalized thermodynamics in his criticism of the Demon. The complexity of his paper and the lack of any accompanying numerical work piqued our curiousity. The internet provides well over two million "hits'' on the subject of "Maxwell's Demon". There are also hundreds of images of the Demon, superimposed upon a container of gas or liquid. However, there is not so much along the lines of simulations of the Demonic process. Accordingly, we thought it useful to write and execute relatively simple FORTRAN programs designed to implement Maxwell's low-density model and to develop its replacement with global Nosé-Hoover or local purely-Newtonian thermal controls. These simulations illustrate the entropy decreases associated with all three types of Demons.

cond-mat.stat-mech↗

arXiv2021

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 {\it irreversible} Second Law of Thermodynamics (Loschmidt's) as well as {\it periodic} 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. The first is time-reversible, but nevertheless dissipative and periodic, the piecewise-linear compressible Baker Map. The fractal properties of that two-dimensional map are mirrored by an even simpler example, the one-dimensional random walk, confined to the unit interval. As a further puzzle the two models yield ambiguities in determining the fractals' information dimensions. These puzzles, including the classical paradoxes, are reviewed and explored here. We review our investigations presented in Budapest in 1997 and end with presentday questions posed as the Snook Prize Problems in 2020 and 2021.

cond-mat.stat-mech↗

arXiv2021

The Simplest Viscous Flow

We illustrate an atomistic periodic two-dimensional stationary shear flow, $u_x = \langle \ \dot x \ \rangle = \dot εy$, using the simplest possible example, the periodic shear of just two particles ! We use a short-ranged "realistic" pair potential, $ϕ(r<2) = (2-r)^6 - 2(2-r)^3$. Many body simulations with it are capable of modelling the gas, liquid, and solid states of matter. A useful mechanics generating steady shear follows from a special ("Kewpie-Doll" $\sim$ "$qp$-Doll") Hamiltonian based on the Hamiltonian coordinates $\{ q \}$ and momenta $\{ p \}$ : ${\cal H}(q,p) \equiv K(p) + Φ(q) + \dot ε\sum qp$. Choosing $qp \rightarrow yp_x$ the resulting motion equations are consistent with steadily shearing periodic boundaries with a strain rate $(du_x/dy) = \dot ε$. The occasional $x$ coordinate jumps associated with periodic boundary crossings in the $y$ direction provide a Hamiltonian that is a piecewise-continuous function of time. A time-periodic isothermal steady state results when the Hamiltonian motion equations are augmented with a continuously variable thermostat generalizing Shuichi Nosé's revolutionary ideas from 1984. The resulting distributions of coordinates and momenta are interesting multifractals, with surprising irreversible consequences from strictly time-reversible motion equations.

nlin.CD↗

arXiv2021

What is Liquid ? [in two dimensions]

We consider the practicalities of defining, simulating, and characterizing "Liquids" from a pedagogical standpoint based on atomistic computer simulations. For simplicity and clarity we study two-dimensional systems throughout. In addition to the infinite-ranged Lennard-Jones 12/6 potential we consider two shorter-ranged families of pair potentials. At zero pressure one of them includes just nearest neighbors. The other longer-ranged family includes twelve additional neighbors. We find that these further neighbors can help stabilize the liquid phase. What about liquids? To implement Wikipedia's definition of liquids as conforming to their container we begin by formulating and imposing smooth-container boundary conditions. To encourage conformation further we add a vertical gravitational field. Gravity helps stabilize the relatively vague liquid-gas interface. Gravity reduces the messiness associated with the curiously-named "spinodal" (tensile) portion of the phase diagram. Our simulations are mainly isothermal. We control the kinetic temperature with Nosé-Hoover thermostating, extracting or injecting heat so as to impose a mean kinetic temperature over time. Our simulations stabilizing density gradients and the temperature provide critical-point estimates fully consistent with previous efforts from free energy and Gibbs' ensemble simulations. This agreement validates our approach.

cond-mat.stat-mech↗

arXiv2020

Time-Symmetry Breaking in Hamiltonian Mechanics. III. A Memoir for Douglas James Henderson [1934-2020]

Following Berni Alder [1] and Francis Ree [2], Douglas Henderson was the third of Bill's California coworkers from the 1960s to die in 2020. Motivated by Doug's death we undertook better to understand Lyapunov instability and the breaking of time symmetry in continuum and atomistic simulations. Here we have chosen to extend our explorations of an interesting pair of nonequilibrium systems, the steady shockwave and the unsteady rarefaction wave. We eliminate the need for boundary potentials by simulating the collisions of pairs of mirror-images projectiles. The resulting shock and rarefaction structures are respectively the results of the compression and the expansion of simple fluids. Shockwaves resulting from compression have a steady structure while the rarefaction fans resulting from free expansions continually broaden. We model these processes using classical molecular dynamics and Eulerian fluid mechanics in two dimensions. Although molecular dynamics is time-reversible the reversed simulation of a steady shockwave compression soon results in an unsteady rarefaction fan, violating the microscopic time symmetry of the motion equations but in agreement with the predictions of macroscopic Navier-Stokes fluid mechanics. The explanations for these results are an interesting combination of two (irreversible) instabilities, Lyapunov and Navier-Stokes.

physics.flu-dyn↗

arXiv2020

Time-Symmetry Breaking in Hamiltonian Mechanics. II. A Memoir for Berni Julian Alder [1925-2020]

This memoir honors the late Berni Julian Alder, who inspired both of us with his pioneering development of molecular dynamics. Berni's work with Tom Wainwright, described in the 1959 Scientific American[1], brought Bill to interview at Livermore in 1962. Hired by Berni, Bill enjoyed over 40 years' research at the Laboratory. Berni, along with Edward Teller, founded UC's Department of Applied Science in 1963. Their motivation was to attract bright students to use the laboratory's unparalleled research facilities. In 1972 Carol was offered a joint LLNL employee-DAS student appointment at Livermore. Bill, thanks to Berni's efforts, was already a Professor at DAS. Carol became one of Bill's best students. Berni's influence was directly responsible for our physics collaboration and our marriage in 1989. The present work is devoted to two early interests of Berni's, irreversibility and shockwaves. Berni and Tom studied the irreversibility of Boltzmann's "H function" in the early 1950s[2]. Berni called shockwaves the "most irreversible" of hydrodynamic processes[3]. Just this past summer, in simulating shockwaves with time-reversible classical mechanics, we found that reversed Runge-Kutta shockwave simulations yielded nonsteady rarefaction waves, not shocks. Intrigued by this unexpected result we studied the exponential Lyapunov instabilities in both wave types. Besides the Runge-Kutta and Leapfrog algorithms, we developed a precisely-reversible manybody algorithm based on trajectory storing, just changing the velocities' signs to generate the reversed trajectories. Both shocks and rarefactions were precisely reversed. Separate simulations, forward and reversed, provide interesting examples of the Lyapunov-unstable symmetry-breaking models supporting the Second Law of Thermodynamics. We describe promising research directions suggested by this work.

physics.hist-ph↗

arXiv2020

Nonequilibrium Molecular Dynamics, Fractal Phase-Space Distributions, the Cantor Set, and Puzzles Involving Information Dimensions for Two Compressible Baker Maps

Deterministic and time-reversible nonequilibrium molecular dynamics simulations typically generate "fractal" [ fractional-dimensional ] phase-space distributions. Because these distributions and their time-reversed twins have zero phase volume, stable attractors "forward in time" and unstable (unobservable) repellors when reversed, these simulations are consistent with the Second Law of Thermodynamics. These same reversibility and stability properties can also be found in compressible Baker Maps, or in their equivalent random walks, motivating their careful study. We illustrate these ideas with three examples: a Cantor-Set Map and two linear compressible Baker Maps, N2$(q,p)$ and N3$(q,p)$. The two Baker Maps' Information dimensions estimated from sequential mappings agree while those from pointwise iteration do not, with the estimates dependent upon details of the approach to the maps' nonequilibrium steady states.

cond-mat.stat-mech↗

arXiv2020

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 which best achieve reproducibility and promote understanding. Few-body models with pair forces have led to today's "realistic" simulations with billions of atoms and molecules. Rapid advances in computer technology have led to change. Theoretical formalisms have largely been replaced by simulations incorporating ingenious algorithm development. We choose to study particularly simple, yet relevant, models directed toward understanding general principles. Simplicity remains a worthy goal, as does relevance. We discuss hard-particle virial series, melting, thermostatted oscillators with and without heat conduction, chaotic dynamics, fractals, the connection of Lyapunov spectra to thermodynamics, and finally simple linear maps. Along the way we mention directions in which additional modelling could provide more clarity and yet more interesting developments in the future.

cond-mat.stat-mech↗

arXiv2020

Compressible Baker Maps and Their Inverses. A Memoir for Francis Hayin Ree [ 1936-2020 ]

This memoir is dedicated to the late Francis Hayin Ree, a formative influence shaping my work in statistical mechanics. Between 1963 and 1968 we collaborated on nine papers published in the Journal of Chemical Physics. Those dealt with the virial series, cell models, and computer simulation. All of them were directed toward understanding the statistical thermodynamics of simple model systems. Our last joint work is also the most cited, with over 1000 citations, "Melting Transition and Communal Entropy for Hard Spheres", submitted 3 May 1968 and published that October. Here I summarize my own most recent work on compressible time-reversible two-dimensional maps. These simplest of model systems are amenable to computer simulation and are providing stimulating and surprising results.

nlin.CD↗

arXiv2019

2020 Ian Snook Prize Problem : Three Routes to the Information Dimensions for a One-Dimensional Stochastic Random Walk and for an Equivalent Prototypical Two-Dimensional Baker Map

The \$1000 Ian Snook Prize for 2020 will be awarded to the author(s) of the most interesting paper exploring a pair of relatively simple, but fractal, models of nonequilibrium systems, a dissipative time-reversible Baker Map and an equivalent stochastic random walk. The two-dimensional deterministic, time-reversible, chaotic, fractal, and dissipative Baker map is equivalent to the stochastic one-dimensional random walk model for which three distinct estimates for the information dimension, $\{ \ 0.7897,\ 0.741_5, \ 0.7337 \ \}$ have all been put forward. So far there is no cogent explanation for the differences among them. We describe the three routes to the information dimension, $D_I$: [ 1 ] iterated Cantor-like mappings, [ 2 ] mesh-based analyses of single-point iterations, and [ 3 ] the Kaplan-Yorke Lyapunov dimension, thought by many to be exact for these models. We encourage colleagues to address this Prize Problem by suggesting, testing, and analyzing mechanisms underlying these differing results.

cond-mat.stat-mech↗

arXiv2019

The Nosé-Hoover, Dettmann, and Hoover-Holian Oscillators

To follow up recent work of Xiao-Song Yang on the Nosé-Hoover oscillator we consider Dettmann's harmonic oscillator, which relates Yang's ideas directly to Hamiltonian mechanics. We also use the Hoover-Holian oscillator to relate our mechanical studies to Gibbs' statistical mechanics. All three oscillators are described by a coordinate $q$ and a momentum $p$. Additional control variables $(ζ, ξ)$ vary the energy. Dettmann's description includes a time-scaling variable $s$, as does Nosé's original work. Time scaling controls the rates at which the $(q,p,ζ)$ variables change. The ergodic Hoover-Holian oscillator provides the stationary Gibbsian probability density for the time-scaling variable $s$. Yang considered {\it qualitative} features of Nosé-Hoover dynamics. He showed that longtime Nosé-Hoover trajectories change energy, repeatedly crossing the $ζ= 0$ plane. We use moments of the motion equations to give two new, different, and brief proofs of Yang's long-time limiting result.

cond-mat.stat-mech↗

arXiv2018

Ergodic Isoenergetic Molecular Dynamics for Microcanonical-Ensemble Averages

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" ( double-well ) potential problem. We consider here a time-reversible dynamical approach to Gibbs' "microcanonical" ( isoenergetic ) distribution for simple systems. To enable isoenergetic ergodicity we add occasional random rotations to the velocities. This idea conserves energy exactly and can be made to cover the entire energy shell with an ergodic dynamics. We entirely avoid the Poincaré-section holes and island chains typical of Hamiltonian chaos. We illustrate this idea for the simplest possible two-dimensional example, a single particle moving in a periodic square-lattice array of scatterers, the "cell model".

cond-mat.stat-mech↗

arXiv2018

Time-Irreversibility is Hidden Within Newtonian Mechanics

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 tension and damped by the viscosities. We characterize the "important" Lyapunov-unstable particles during the collision and equilibration phases in both time directions to demonstrate the utility of the Milne algorithm in exposing "Time's Arrow".

cond-mat.stat-mech↗