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Computational Physics with Particles – Nonequilibrium Molecular Dynamics and Smooth Particle Applied Mechanics

100%

Hoover W. G.

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tom Vol. 13, nr 2

83-93

EN

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.

2

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

63%

Hoover W. G. , Hoover C. G.

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tom 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.

3

2019 SNOOK Prizes in Computational Statistical Mechanics

63%

Hoover W. G. , Hoover C. G.

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tom 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.

4

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

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Hoover W. G. , Hoover C. G.

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tom 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.

5

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

63%

Hoover W. G. , Hoover C. G.

Computational Methods in Science and Technology

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2017

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tom 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.

6

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

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Hoover C. G. , Hoover W. G.

Computational Methods in Science and Technology

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2017

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tom 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.

7

Time - Symmetry Breaking in Hamiltonian Mechanics

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Hoover W. G. , Hoover C. G.

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tom 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.

8

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

63%

Hoover W. G. , Hoover C. G.

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tom 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.

9

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

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Hoover W. G. , Hoover C. G.

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tom 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.

10

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

63%

Hoover W. G. , Hoover C. G.

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tom 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.

11

Dynamical instabilities, manifolds, and local Lyapunov spectra far from equilibrium

51%

Hoover W. G. , Hoover C. G. , Posch H. A.

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tom Vol. 7, nr 1

55-65

EN

We consider an harmonic oscillator in a thermal gradient far from equilibrium. The motion is made ergodic and fully time-reversible through the use of two thermostat variables. The equations of motion contain both linear and quadratic terms. The dynamics is chaotic. The resulting phase-space distribution is not only complex and multifractal, but also ergodic, due to the time-reversibility property. We analyze dynamical time series in two ways. We describe local, but comoving, singularities in terms of the "local Lyapunov spectrum" {gamma}. Local singularities at a particular phase-space point can alternatively be described by the local eigenvalues and eigenvectors of the "dynamical matrix"D identical to partial differential ni/partial differential r identical to Nabla ni. D is the matrix of derivatives of the equations of motion r=ni(r) . We pursue this eigenvalue-eigenvector description for the oscillator. We find that it breaks down at a dense set of singular points, where the four eigenvectors span only a three-dimensional subspace. We believe that the concepts of stable and unstable global manifolds are problematic for this simple nonequilibrium system. PACS numbers: 05.45.A, 05.10, 07.05.T