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In 1984 Shuichi Nosé invented an isothermal mechanics designed to generate Gibbs' canonical distribution for the coordinates {q} and momenta {p} of classical N-body systems [1, 2]. His approach introduced an additional timescaling variable s that could speed up or slow down the {q, p} motion in such a way as to generate the Gaussian velocity distribution ∝ e-p2/2mkT and the corresponding potential distribution, ∝ e-F(q)/kT. (For convenience here we choose Boltzmann's constant k and the particle massmboth equal to unity.) SoonWilliam Hoover pointed out that Nosé's approach fails for the simple harmonic oscillator [3]. Rather than generating the entire Gaussian canonical oscillator distribution, the Nosé-Hoover approach, which includes an additional friction coefficient ζ with distribution e-ζ2/2/v 2p, generates only a modest fractal chaotic sea, filling a small percentage of the canonical (q, p, ζ) distribution. In the decade that followed this thermostatted work a handful of ergodic algorithms were developed in both three- and four-dimensional phase spaces. These new approaches generated the entire canonical distribution, without holes. The 2024 Snook Prize problem is to study the efficiency of several such algorithms, such as the five ergodic examples described here, so as to assess their relative usefulness in attaining the canonical steady state for the harmonic oscillator. The 2024 Prize rewarding the best assessment is United States $1000, half of it a gift from ourselves with the balance from the Pozna'n Supercomputing and Networking Center. [ABSTRACT FROM AUTHOR]

Baker Maps have long served as pedagogical tools for understanding chaos and fractal phase-space distributions. Recent work [1], following earlier efforts from 1997 [2], shows that the Kaplan-Yorke formula for information dimension disagrees with direct computation for some simple compressible Baker Maps. Here we extend this map work to simple continuous flows. We compare pointwise information dimensions to the Kaplan-Yorke dimension for a simple four-dimensional flow [3] controlling both ⟨p4⟩ and ⟨p2⟩: { q˙ = p; p˙ = −q − ξp3 − ζp; ξ˙ = p4 − 3p2; ζ˙ = p2 − T }. Precisely similar sets of Gaussian points could be generated with Metropolis’ Monte-Carlo simulations of harmonic oscillators in Gibbs’ canonical ensemble with f(q) = e−q2/2/p (2π). Remarkably, we show that the dependence of the pointwise information dimension for the Gaussian distribution is linear in the inverse of the logarithm of the mesh spacing, ∝ 1/ln(1/δ). The Hoover-Holian Gaussian oscillator problem [3] can be generalized [2–4] to some nonequilibrium steady-state problems by introducing a temperature-gradient parameter ϵ. In that case the temperature T varies from 1 − ϵ to 1 + ϵ: T = 1 + ϵ tanh(q) so that both conservative (ϵ = 0) and dissipative (ϵ > 0) flows result. [ABSTRACT FROM AUTHOR]

25 years ago the June 1998 Focus Issue of "Chaos" described the proceedings of a workshop meeting held in Budapest and called "Chaos and Irreversibility", by the organizers, T. Tél, P. Gaspard, and G. Nicolis. These editors organized the meeting and the proceedings' issue. They emphasized the importance of fractal structures and Lyapunov instability to modelling nonequilibrium steady states. Several papers concerning maps were presented. Ronald Fox considered the entropy of the incompressible Baker Map B(x, y), shown here in Fig. 1. He found that the limiting probability density after many applications of the map is ambiguous, depending upon the way the limit is approached. Harald Posch and Bill Hoover considered a time-reversible version of a compressible Baker Map, with the compressibility modelling thermostatting. Now, 25 years later, we have uncovered a similar ambiguity, with the information dimension of the probability density giving one value from pointwise averaging and a different one with areawise averaging. Goldstein, Lebowitz, and Sinai appear to consider similar ambiguities. Tasaki, Gilbert, and Dorfman note that the Baker Map probability density is singular everywhere, though integrable over the fractal y coordinate. Breymann, Tél, and Vollmer considered the concatenation of Baker Maps into MultiBaker Maps, as a step toward measuring spatial transport with dynamical systems. The present authors have worked on Baker Maps ever since the 1997 Budapest meeting described in "Chaos". This paper provides a number of computational benchmark simulations of "Generalized Baker Maps" (where the compressibility of the Map is varied or "generalized") as described by Kumicák in 2005. [ABSTRACT FROM AUTHOR]

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, ∝ eλt where λ is a “Lyapunov exponent”. [ABSTRACT FROM AUTHOR]

Carnot's four-part ideal-gas cycle includes both isothermal and adiabatic expansions and compressions. Analyzing this cycle provides the fundamental basis for statistical thermodynamics. We explore the cycle here from a pedagogical view in order to promote understanding of the macroscopic thermodynamic entropy, the state function associated with thermal energy changes. From the alternative microscopic viewpoint the Hamiltonian H(q, p) is the energy and entropy is the (logarithm of the) phase-space volume O associated with a macroscopic state. We apply two novel forms of Hamiltonian mechanics to Carnot's Cycle: (1) Gauss' isokinetic mechanics for the isothermal segments and (2) Doll's Tensor mechanics for the isentropic adiabatic segments. We explore the equivalence of the microscopic and macroscopic views of Carnot's cycle for simple fluids here, beginning with the ideal Knudsen gas and extending the analysis to a prototypical simple fluid. [ABSTRACT FROM AUTHOR]

An interesting preprint by Puru Gujrati, "Maxwell's Demon Must Remain Subservient to Clausius' Statement" [of the Second Law of Thermodynamics], traces the development and application of Maxwell's Demon. He argues against the Demon on thermodynamic grounds. Gujrati introduces and uses his own version of a generalized thermodynamics in his criticism of the Demon. The complexity of his paper and the lack of any accompanying numerical work piqued our curiousity. The internet provides well over two million "hits" on the subject of "Maxwell's Demon". There are also hundreds of images of the Demon, superimposed upon a container of gas or liquid. However, there is not so much along the lines of simulations of the Demonic process. Accordingly, we thought it useful to write and execute relatively simple FORTRAN programs designed to implement Maxwell's low-density model and to develop its replacement with global Nosé-Hoover or local purely-Newtonian thermal controls. These simulations illustrate the entropy decreases associated with all three types of Demons. [ABSTRACT FROM AUTHOR]

We illustrate an atomistic periodic two-dimensional stationary shear flow, ux = h = έy, 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) + ø (q) + έ ∑ qp. Choosing qp → ypx the resulting motion equations are consistent with steadily shearing periodic boundaries with a strain rate (dux/dy) = έ. 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Typical Hamiltonian liquids display exponential "Lyapunov instability", also called "sensitive dependence on initial conditions". Although Hamilton's equations are thoroughly time-reversible, the forward and backward Lyapunov instabilities can differ, qualitatively. In numerical work, the expected forward/backward pairing of Lyapunov exponents is also occasionally violated. To illustrate, we consider many-body inelastic collisions in two space dimensions. Two mirror-image colliding crystallites can either bounce, or not, giving rise to a single liquid drop, or to several smaller droplets, depending upon the initial kinetic energy and the interparticle forces. The difference between the forward and backward evolutionary instabilities of these problems can be correlated with dissipation and with the Second Law of Thermodynamics. Accordingly, these asymmetric stabilities of Hamilton's equations can provide an "Arrow of Time". We illustrate these facts for two small crystallites colliding so as to make a warm liquid. We use a specially-symmetrized form of Levesque and Verlet's bit-reversible Leapfrog integrator. We analyze trajectories over millions of collisions with several equally-spaced time reversals. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The microscopic and macroscopic versions of fluid mechanics differ qualitatively. Microscopic particles obey time-reversible ordinary differential equations. The resulting particle trajectories {q(t)g} may be time-averaged or ensemble-averaged so as to generate field quantities corresponding to macroscopic variables. On the other hand, the macroscopic continuum fields described by fluid mechanics follow irreversible partial differential equations. Smooth particle methods bridge the gap separating these two views of fluids by solving the macroscopic field equations with particle dynamics that resemble molecular dynamics. Recently, nonlinear dynamics have provided some useful tools for understanding the relationship between the microscopic and macroscopic points of view. Chaos and fractals play key roles in this new understanding. Non-equilibrium phase-space averages look very different from their equilibrium counterparts. Away from equilibrium the smooth phase-space distributions are replaced by fractional-dimensional singular distributions that exhibit time irreversibility. [ABSTRACT FROM AUTHOR]

The authors thermostat a qp harmonic oscillator using the two additional control variables zeta and xi to simulate Gibbs' canonical distribution. In contrast to the motion of purely Hamiltonian systems, the thermostated oscillator motion is completely ergodic, covering the full four-dimensional [q,p,zeta,xi] phase space. The local Lyapunov spectrum (instantaneous growth rates of a comoving corotating phase-space hypersphere) exhibits singularities like those found earlier for Hamiltonian chaos, reinforcing the notion that chaos requires kinetic-as opposed to statistical-study, both at and away from equilibrium. The exponent singularities appear to have a fractal character.