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What is Liquid? [in two dimensions]

Travis K. P., Hoover Wm. G., Hoover C. G., Hass A. B.

Computational Methods in Science and Technology

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2021

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Vol. 27, No. 1

5--23

EN

We consider the practicalities of defining, simulating, and characterizing “Liquids” from a pedagogical standpoint based on atomistic computer simulations. For simplicity and clarity we study two-dimensional systems throughout. In addition to the infinite-ranged Lennard-Jones 12/6 potential we consider two shorter-ranged families of pair potentials. At zero pressure one of them includes just nearest neighbors. The other longer-ranged family includes twelve additional neighbors. We find that these further neighbors can help stabilize the liquid phase. What about liquids? To implement Wikipedia’s definition of liquids as conforming to their container we begin by formulating and imposing smooth-container boundary conditions. To encourage conformation further we add a vertical gravitational field. Gravity helps stabilize the relatively vague liquid-gas interface. Gravity reduces the messiness associated with the curiously-named “spinodal” (tensile) portion of the phase diagram. Our simulations are mainly isothermal. We control the kinetic temperature with Nosé-Hoover thermostating, extracting or injecting heat so as to impose a mean kinetic temperature over time. Our simulations stabilizing density gradients and the temperature provide critical-point estimates fully consistent with previous efforts from free energy and Gibbs ensemble simulations. This agreement validates our approach.

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The Simplest Viscous Flow

Hoover Wm. G., Hoover C. G.

Computational Methods in Science and Technology

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2021

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Vol. 27, No. 3

109--119

EN

We illustrate an atomistic periodic two-dimensional stationary shear flow, ux = h x˙ i = ˙y, using the simplest possible example, the periodic shear of just two particles! We use a short-ranged “realistic” pair potential, φ(r < 2) = = (2 − r) 6 − 2(2 − r) 3 . Many body simulations with it are capable of modelling the gas, liquid, and solid states of matter. A useful mechanics generating steady shear follows from a special (“Kewpie-Doll” ∼ “qp-Doll”) Hamiltonian based on the Hamiltonian coordinates {q} and momenta {p} : H(q, p) ≡ K(p) + Φ(q) + ˙ Pqp. Choosing qp → ypx the resulting motion equations are consistent with steadily shearing periodic boundaries with a strain rate (dux/dy) = ˙. The occasional x coordinate jumps associated with periodic boundary crossings in the y direction provide a Hamiltonian that is a piecewise-continuous function of time. A time-periodic isothermal steady state results when the Hamiltonian motion equations are augmented with a continuously variable thermostat generalizing Shuichi Nosé’s revolutionary ideas from 1984. The resulting distributions of coordinates and momenta are interesting multifractals, with surprising irreversible consequences from strictly time-reversible motion equations.

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$1000 SNOOK PRIZES FOR 2021 : The Information Dimensions of a Two-Dimensional Baker Map

Hoover Wm. G., Hoover C. G.

Computational Methods in Science and Technology

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2021

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Vol. 27, No. 2

93--94

EN

The fractal information dimension can be computed in three ways: (1) mapping points, (2) mapping regions (two-dimensional areas here), and (3) applying the Kaplan-Yorke conjecture. For the simplest nonequilibrium Baker N2 Map these three approaches can give different results. A pedagogical exploration and explanation of this situation is the 2021 Ian Snook Prize Problem.

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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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Compressible Baker Maps and Their Inverses : A Memoir for Francis Hayin Ree [1936–2020]

Hoover Wm. G.

Computational Methods in Science and Technology

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2020

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Vol. 26, No. 1

5--13

EN

This memoir is dedicated to the late Francis Hayin Ree, a formative influence shaping my work in statistical mechanics. Between 1963 and 1968 we collaborated on nine papers published in the Journal of Chemical Physics. Those dealt with the virial series, cell models, and computer simulation. All of them were directed toward understanding the statistical thermodynamics of simple model systems. Our last joint work is also the most cited, with over 1000 citations, “Melting Transition and Communal Entropy for Hard Spheres”, submitted 3 May 1968 and published that October. Here I summarize my own most recent work on compressible time-reversible two-dimensional maps. These simplest of model systems are amenable to computer simulation and are providing stimulating and surprising results.