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

Hoover W. G., Sprott J. C., Hoover C. G.

Computational Methods in Science and Technology

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2019

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Vol. 25, No. 3

121--124

EN

To follow up recent work of Xiao-Song Yang [1] on the Nosé-Hoover oscillator [2–5] we consider Dettmann’s harmonic oscillator [6, 7], which relates Yang’s ideas directly to Hamiltonian mechanics. We also use the Hoover-Holian oscillator [8] 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 [2, 3]. 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 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.

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Aspects of Dynamical Simulations, Emphasizing Nosé and Nosé-Hoover Dynamics and the Compressible Baker Map

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2019

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Vol. 25, No. 3

125--141

EN

Aspects of the Nosé and Nosé-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. Nosé, Nosé-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 phasespace 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.

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2020 Ian Snook Prize Problem : Three Routes to the Information Dimensions for One-Dimensional Stochastic Random Walks and Their Equivalent Two-Dimensional Baker Maps

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2019

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Vol. 25, No. 4

153--159

EN

The $1000 Ian Snook Prize for 2020 will be awarded to the author(s) of the most interesting paper exploring pairs of relatively simple, but fractal, models of nonequilibrium systems, dissipative time-reversible Baker Maps and their equivalent stochastic random walks. Two-dimensional deterministic, time-reversible, chaotic, fractal, and dissipative Baker maps are equivalent to stochastic one-dimensional random walks. Three distinct estimates for the information dimension, {0.7897, 0.7415, 0.7337} have all been put forward for one such model. So far there is no cogent explanation for the differences among these estimates. We describe the three routes to the information dimension, DI : 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.

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2019 SNOOK Prizes in Computational Statistical Mechanics

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2019

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Vol. 25, No. 1

5

EN

The one-dimensional φ 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, φ 4 chains exhibit Fourier heat conduction. In his recent Snook-Prize work Aoki also showed that the model can exhibit chaos on the threedimensional energy surface describing a two-body two-spring chain. That surface can include at least two distinct chaotic seas. Aoki pointed out that the model typically exhibits different kinetic temperatures for the two bodies. Evidently few-body φ 4 problems merit more investigation. Accordingly, the 2019 Prizes honoring Ian Snook (1945–2013) [five hundred United States dollars cash from the Hoovers and an additional $500 cash from the Institute of Bioorganic Chemistry of the Polish Academy of Sciences and the Poznan Supercomputing and Networking Center] will be awarded to the author(s) of the most interesting work analyzing and discussing few-body φ 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.

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The Φ4 Model, Chaos, Thermodynamics, and the 2018 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

159--163

EN

The one-dimensional Φ4 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, Φ4 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 a two-body two-spring chain. That surface can include at least two distinct chaotic seas. Aoki pointed out that the model typically exhibits different kinetic temperatures for the two bodies. Evidently few-body Φ4 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 Φ4 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.

6

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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Ergodic Isoenergetic Molecular Dynamics for Microcanonical-Ensemble Averages

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

155--158

EN

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” (doublewell) 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”.

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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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Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize Award

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 1

5--8

EN

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 three one-dimensional problems, the harmonic oscillator, the quartic oscillator, and the Mexican Hat potentials: {(q2=2); (q4=4); (q4=4) 􀀀 (q2=2)}. Their work constitutes an effective response to the 2016 Ian Snook Prize Award goal, “finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat”. We confirm their work here and highlight an interesting feature of the Mexican Hat problem when it is solved with an adaptive integrator.

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Instantaneous Pairing of Lyapunov Exponents in Chaotic Hamiltonian Dynamics and the 2017 Ian Snook Prizes

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2017

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Vol. 23, No. 1

73--79

EN

The time-averaged Lyapunov exponents, f_ig, 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, γ1 + γ2, describe the rate for areas defined by three nearby trajectories. γ1 + γ2 + γ3 is the rate for volumes defined by four nearby trajectories, and so on. Lyapunov exponents for Hamiltonian systems are symmetric. The time-reversibility of the motion equations links the growth and decay rates together in pairs. This pairing provides a more detailed explanation than Liouville’s for the conservation of phase volume in Hamiltonian mechanics. Although correct for long-time averages, the dependence of trajectories on their past is responsible for the observed lack of detailed pairing for the instantaneous “local” exponents, {γi(t)}. The 2017 Ian Snook Prizes will be awarded to the author(s) of an accessible and pedagogical discussion of local Lyapunov instability in small systems. We desire that this discussion build on the two nonlinear models described here, a double pendulum with Hooke’s-Law links and a periodic chain of Hooke’s-Law particles tethered to their lattice sites. The latter system is the ϕ4 model popularized by Aoki and Kusnezov. A four-particle version is small enough for comprehensive numerical work and large enough to illustrate ideas of general validity.

11

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.

13

Singly-Thermostated Ergodicity in Gibbs’ Canonical Ensemble and the 2016 Ian Snook Prize

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2016

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Vol. 22, No. 3

127--131

EN

For a harmonic oscillator, Nosé’s single-thermostat approach to simulating Gibbs’ canonical ensemble with dynamics samples only a small fraction of the phase space. Nosé’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 ergodic fashion; [2] sets of single-thermostat motion equations, exerting “weak control” over both forces and momenta, have been shown to be ergodic; and [3] sets of single-thermostat motion equations exerting weak control over two velocity moments provide ergodic phase-space sampling for the oscillator and for the rigid pendulum, but not for the quartic oscillator or for the Mexican Hat potential. The missing fourth step, motion equations providing ergodic sampling for anharmonic potentials requires a further advance. The 2016 Ian Snook Prize will be awarded to the author(s) of the most interesting original submission addressing the problem of finding ergodic algorithms for Gibbs’ canonical ensemble using a single thermostat.

14

Ian Snook Prizes 2015

Hoover W. G., Hoover C. G., Stroiński M., Węglarz J., Wojciechowski K. W.

Computational Methods in Science and Technology

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2016

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Vol. 22, No. 2

59

15

Time-Reversible Ergodic Maps and the 2015 Ian Snook Prizes

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

161--164

EN

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)↔(q;--p). Weak control of the temperature, α p2 and its fluctuation, resulting in ergodicity, has recently been achieved in a threedimensional time-reversible model of a heat-conducting harmonic oscillator. Two-dimensional cross sections of such nonequilibrium flows can be generated with time-reversible dissipative maps yielding æsthetically interesting attractorrepellor pairs. We challenge the reader to find and explore such time-reversible dissipative maps. This challenge is the 2015 Snook-Prize Problem.

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Ergodicity of the Martyna-Klein-Tuckerman Thermostat and the 2014 Ian Snook Prize

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2015

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Vol. 21, No. 1

5--10

EN

Nosé and Hoover’s 1984 work showed that although Nosé and Nosé-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, to generate Gibbs’ canonical distribution for an oscillator. Three successful time-reversible and deterministic sets of twothermostat motion equations were developed in the 1990s. We analyze one of them here. It was developed by Martyna, Klein, and Tuckerman in 1992. Its ergodicity was called into question by Patra and Bhattacharya in 2014. This question became the subject of the 2014 Snook Prize. Here we summarize the previous work on this problem and elucidate new details of the chaotic dynamics in the neighborhood of the two fixed points. We apply six separate tests for ergodicity and conclude that the MKT equations are fully compatible with all of them, in consonance with our recent work with Clint Sprott and Puneet Patra.

17

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.

18

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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Ergodicity of a Time-Reversibly Thermostated Harmonic Oscillator and the 2014 Ian Snook Prize

Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2014

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Vol. 20, No. 3

87--92

EN

Shuichi Nosé 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. Nosé’s dynamical Hamiltonian bridge requires the “ergodic” support of a space-filling structure in order to reproduce the entire distribution. For sufficiently small systems, such as the harmonic oscillator, Nosé’s dynamical approach failed to agree with Gibbs’ sampling and instead showed a complex structure, partitioned into a chaotic sea, islands, and chains of islands, that is familiar textbook fare from investigations of Hamiltonian chaos. In trying to enhance small-system ergodicity several more complicated “thermostated" equations of motion were developed. All were consistent with the canonical Gaussian distribution for the oscillator coordinate and momentum. The ergodicity of the various approaches has undergone several investigations, with somewhat inconclusive (contradictory) results. Here we illustrate several ways to test ergodicity and challenge the reader to find even more convincing algorithms or an entirely new approach to this problem.

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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.