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2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

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2023

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Vol. 29, No. 1-4

65--69

EN

In 1984 Shuichi Nosé invented an isothermal mechanics designed to generate Gibbs’ canonical distribution for the coordinates {q} and momenta {p} of classical N-body systems [1, 2]. His approach introduced an additional timescaling variable s that could speed up or slow down the {q, p} motion in such a way as to generate the Gaussian velocity distribution ∝ e −p 2/2mkT and the corresponding potential distribution, ∝ e −Φ(q)/kT . (For convenience here we choose Boltzmann’s constant k and the particle mass m both equal to unity.) Soon William Hoover pointed out that Nosé’s approach fails for the simple harmonic oscillator [3]. Rather than generating the entire Gaussian canonical oscillator distribution, the Nosé-Hoover approach, which includes an additional friction coefficient ζ with distribution e −ζ 2/2 / √ 2π, generates only a modest fractal chaotic sea, filling a small percentage of the canonical (q, p, ζ) distribution. In the decade that followed this thermostatted work a handful of ergodic algorithms were developed in both three- and four-dimensional phase spaces. These new approaches generated the entire canonical distribution, without holes. The 2024 Snook Prize problem is to study the efficiency of several such algorithms, such as the five ergodic examples described here, so as to assess their relative usefulness in attaining the canonical steady state for the harmonic oscillator. The 2024 Prize rewarding the best assessment is United States $1000, half of it a gift from ourselves with the balance from the Poznan Supercomputing ´ and Networking Center.

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Time-symmetry breaking in Hamiltonian mechanics. Part 3, A memoir for Douglas James Henderson [1934–2020]

Hoover Wm. G., Hoover C. G.

Computational Methods in Science and Technology

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2020

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Vol. 26, No. 4

111--120

EN

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 [1, 2]. 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.

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Time-symmetry breaking in Hamiltonian mechanics. Part 2, A memoir for Berni Julian Alder [1925–2020],

Hoover Wm. G., Hoover C. G.

Computational Methods in Science and Technology

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2020

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Vol. 26, No. 4

101--110

EN

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 there. 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.

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The 2017 SNOOK PRIZES in Computational Statistical Mechanics

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2018

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Vol. 24, No. 2

75--79

EN

The 2017 Snook Prize has been awarded to Kenichiro Aoki for his exploration of chaos in Hamiltonian 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 Hénon-Heiles models and has been selected for Honorable Mention in the Snook-Prize competition.

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Yokohama to Ruby Valley : Around the World in 80 Years. II.

Hoover C. G., Hoover W. G.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 3

143--153

EN

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 in Nevada in 2005, which has turned out to be a long-term working vacation. Carol narrates this part of our history together.

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From Ann Arbor to Sheffield : Around the World in 80 Years. I.

Hoover W. G.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 3

133--141

EN

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 Birthday celebration at the University of Sheffield in July 2016.

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Bit-Reversible Version of Milne’s Fourth-Order Time-Reversible Integrator for Molecular Dynamics

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 4

299--303

EN

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 = (q2 + v2)=2. (We use this one-dimensional oscillator problem as an illustration throughout this paper). Milne’s integrator is particularly useful for the analysis of Lyapunov (exponential) instability in dynamical systems, including manybody molecular dynamics. We include the details necessary to the implementation of Milne’s Algorithms.

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Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2015

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Vol. 21, No. 3

109--116

EN

Time-reversible 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 apparent advantage of such methods. But all numerical methods are susceptible to Lyapunov instability, which severely limits the maximum time for which chaotic solutions can be “accurate”. The “advantages” of higher-order methods are lost rapidly for typical chaotic Hamiltonians. We illustrate these difficulties for a useful reproducible test case, the twodimensional one-particle cell model with specially smooth forces. This Hamiltonian problem is chaotic and occurs on a three-dimensional constant-energy shell, the minimum dimension for chaos. We benchmark the problem with quadrupleprecision trajectories using the fourth-order Candy-Rozmus, fifth-order Runge-Kutta, and eighth-order Schlier-Seiter-Teloy integrators. We compare the last, most-accurate particle trajectories to those from six double-precision algorithms, four symplectic and two Runge-Kutta.

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Why Instantaneous Values of the “Covariant” Lyapunov Exponents Depend upon the Chosen State-Space Scale

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2014

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Vol. 20, No. 1

5--8

EN

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 ζ(t) is unchanged by such scaling. The original (qpζ) motion and the scale-model (QPζ) 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.

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Time’s Arrow for Shockwaves; Bit - Reversible Lyapunov and “Covariant” Vectors; Symmetry Breaking

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2013

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Vol. 19, No. 2

69--75

EN

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 nearby “satellite” trajectories. From the forward and backward processes we 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 timesymmetry 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.

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Time - Symmetry Breaking in Hamiltonian Mechanics

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2013

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Vol. 19, No. 2

77--87

EN

Hamiltonian trajectories are strictly time-reversible. Any time series of Hamiltonian coordinates f q g satisfying Hamilton’s motion equations will likewise satisfy them when played “backwards”, with the corresponding momenta changing signs: {+p }→{-g}. 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 ≃ eλt. Surprisingly, the positive and negative exponents, as well as the forward and backward Lyapunov spectra , {λforward(t) } and {λt backward(t) }, 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, [i] a minimal (but still chaotic) one-body “cell model” with a four-dimensional phase space; [ii] relatively small colliding crystallites, for which the whole Lyapunov spectrum is accessible; [iii] 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.

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Erratum: Time’s Arrow for Shockwaves; Bit-Reversible Lyapunov and “Covariant” Vectors; Symmetry Breaking

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2013

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Vol. 19, No. 3

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