Wyniki wyszukiwania - BazTech

1

Information Dimensions of Simple Four-Dimensional Flows

Hoover William Graham, Hoover Carol Griswold, Travis Karl Patrick

Computational Methods in Science and Technology

|

2023

|

Vol. 29, No. 1-4

21--26

EN

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 ⟨p 4 ⟩ and ⟨p 2 ⟩: { q˙ = p ; ˙p = −q − ξp3 − ζp ; ˙ξ = p 4 − 3p 2 ; ˙ζ = p 2 − 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 −q 2/2 / 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.

2

A Quarter Century of Baker-Map Exploration

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

|

2023

|

Vol. 29, No. 1-4

5--16

EN

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.

3

2024 Snook Prize Problem: Ergodic Algorithms’ Mixing Rates

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

|

2023

|

Vol. 29, No. 1-4

65--69

EN

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 −p 2/2mkT and the corresponding potential distribution, ∝ e −Φ(q)/kT . (For convenience here we choose Boltzmann’s constant k and the particle mass m both equal to unity.) Soon William 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 −ζ 2/2 / √ 2π, 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 Poznan Supercomputing ´ and Networking Center.

4

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

|

2022

|

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”.

5

Thermodynamic Entropy from Sadi Carnot’s Cycle using Gauss’ and Doll’s-Tensor Molecular Dynamics

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

|

2022

|

Vol. 28, No. 2

41--46

EN

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 Ω 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.

6

Computational Implementation of Maxwell’s Knudsen-Gas Demon and Its Extension to a Two-Dimensional Soft-Disk Fluid

Hoover William Graham, Hoover Carol Griswold

Computational Methods in Science and Technology

|

2022

|

Vol. 28, No. 1

5--9

EN

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