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

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

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2022

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Vol. 28, No. 4

133--141

EN

We analyze the time-reversible mechanics of two irreversible simulation types. The first is a dissipative onedimensional 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. This instability results in the exponential growth of small perturbations, ∝ e λt where λ is a “Lyapunov exponent”.

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Two-dimensional Time-reversible Ergodic Maps with Provisions for Dissipation

Patra P. K.

Computational Methods in Science and Technology

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2016

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Vol. 22, No. 2

61--70

EN

A new discrete time-reversible map of a unit square onto itself is proposed. The map comprises of piecewise linear two-dimensional operations, and is able to represent the macroscopic features of both equilibrium and nonequilibrium dynamical systems. Our operations are analogous to sinusoidally driven shear in the two dimensions, and a radial compression/ expansion of a point lying outside/inside a circle centred around origin. Depending upon the radius, the map transitions from being ergodic and non-dissipative (like in equilibrium situations) to a limit cycle through intermediate multifractal situations (like in nonequilibrium situations). All dissipative cases of the proposed map suggest that the Kaplan-Yorke dimension is smaller than the embedding dimension, a feature typically arising in nonequilibrium steady-states. The proposed map differs from the existing maps like the Baker map and Arnold’s cat map in the sense that (i) it is reversible, and (ii) it generates an intricate multifractal phase-space portrait.

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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.

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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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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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