Nonequilibrium Molecular Dynamics, Fractal Phase-Space Distributions, the Cantor Set, and Puzzles Involving Information
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Nonequilibrium Molecular Dynamics, Fractal Phase-Space Distributions, the Cantor Set, and Puzzles Involving Information Dimensions for Two Compressible Baker Maps William G. Hoover1* and 1
Carol G. Hoover1**
Ruby Valley Research Institute, Highway Contract 60, Box 601 Ruby Valley, 89833 NV, USA
Received April 26, 2020; revised July 20, 2020; accepted July 31, 2020
Abstract—Deterministic and time-reversible nonequilibrium molecular dynamics simulations typically generate “fractal” (fractional-dimensional) phase-space distributions. Because these distributions and their time-reversed twins have zero phase volume, stable attractors “forward in time” and unstable (unobservable) repellors when reversed, these simulations are consistent with the second law of thermodynamics. These same reversibility and stability properties can also be found in compressible baker maps, or in their equivalent random walks, motivating their careful study. We illustrate these ideas with three examples: a Cantor set map and two linear compressible baker maps, N2(q, p) and N3(q, p). The two baker maps’ information dimensions estimated from sequential mappings agree, while those from pointwise iteration do not, with the estimates dependent upon details of the approach to the maps’ nonequilibrium steady states. MSC2010 numbers: 01-08, 34-03, 70H14, 82C05 DOI: 10.1134/S1560354720050020 Keywords: chaos, Lyapunov exponents, irreversibility, random walks, maps, information dimension
1. NONEQUILIBRIUM MOLECULAR DYNAMICS GENERATES FRACTALS The computers developed for the National Laboratories were first applied to manybody problems in the 1950s. At Los Alamos in 1953, Fermi, Pasta, and Ulam [1] described the incomplete equilibration of one-dimensional waves in anharmonic chains. Soon afterward, at Livermore, Berni Alder and Tom Wainwright simulated the motion of systems of several hundred hard disks and spheres [2]. At Brookhaven George Vineyard and his coworkers studied “realistic” atomistic models of the impact of high-energy radiation on models of simple metals shortly thereafter [3]. All of these atomistic simulations were developed based on classical Newtonian mechanics with shortranged pairwise-additive forces. “Large” simulations involved several hundred discrete particles. A generation later simulations with millions of particles were possible. Figure 1 shows a typical simulation from our 1989–1990 visit to Japan. These indentations of amorphous Stillinger – Weber silicon, using two different indentor models, generate plastic flow near the indentors [4]. Trillionatom simulations are feasible in 2020. In 1984 Shuichi Nos´e had announced a revolutionary method for imposing specified temperatures and pressures on molecular dynamics simulations [5, 6]. His modification of Hamiltonian mechanics was designed to replicate Gibbs’ isothermal and isobaric ensembles. Equilibrium distributions had been formulated by Gibbs’ statistical mechanics prior to the close of the 19th century. To match Gibbs’ results Nos´e found it necessary to introduce a “scaled”
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