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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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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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Maxwell and Cattaneo’s Time - Delay Ideas Applied to Shockwaves and the Rayleigh - Bénard Problem

Uribe F. J., Hoover W. G., Hoover C. G.

Computational Methods in Science and Technology

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2013

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Vol. 19, No. 1

5--12

EN

We apply Maxwell and Cattaneo’s relaxation approaches to the analysis of strong shockwaves in a two-dimensional viscous heat-conducting fluid. Good agreement results for reasonable values of Maxwell’s relaxation times. Instability results if the viscous relaxation time is too large. These relaxation terms have negligible effects on slower-paced subsonic problems, as is shown here for two-roll and four-roll Rayleigh-Bénard flow

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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

e1

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Modelowanie propagacji powybuchowych fal uderzeniowych w przestrzeni ograniczonej dwiema sztywnymi płaszczyznami

Borkowski W., Rafa J., Łęgowski Z., Rybak P.

Biuletyn Wojskowej Akademii Technicznej

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1998

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Vol. 47, nr 11

73-98

PL

Opracowano metodę szacowania obciążeń płyty generowanych wybuchem kontaktowym skondensowanego materiału wybuchowego. Problem propagacji powybuchowej fali uderzeniowej w przestrzeni ograniczonej dwiema sztywnymi płaszczyznami zredukowano do rozwiązania trzech zagadnień cząstkowych: rozwiązania problemu propagacji kulistej fali uderzeniowej wybuchu w nieograniczonym ośrodku gazowym, opracowania lokalnej teorii odbicia skośnej fali uderzeniowej od sztywnej ścianki oraz zbudowania efektywnej teorii opisującej propagację układu odbitych fal uderzeniowych przy wykorzystaniu odpowiedniej modyfikacji metody źródeł pozornych. Uzyskano dobrą zgodność obliczeń teoretycznych z wynikami przeprowadzonych eksperymentów.

EN

A method of estimating the rigid surface loading generated by the contract explosion of a condensed explosive charge has been elaborated. The problem of propagation of an explosive generated shock wave in the zone bounded by two rigid plates has been reduced to solving three subproblems: the propagation problem of spherical shock wave in an unbounded gaseous medium, working out the local theory of an oblique shock wavereflection at a rigid wall and constructing an effective theory describing propagation of the reflected shock waves system, making use of an appropriate modyfication of the dummy sources method. A good agreement has been obtained of the results of theoretical computations with performed experimental measurements.