The Φ4 Model, Chaos, Thermodynamics, and the 2018 SNOOK Prizes in Computational Statistical Mechanics

• Autorzy: Hoover W. G. , Hoover C. G.

Identyfikatory

DOI

10.12921/cmst.2018.0000032

• Języki publikacji: EN

Abstrakty

EN

The one-dimensional Φ4 Model generalizes a harmonic chain with nearest-neighbor Hooke’s-Law interactions by adding quartic potentials tethering each particle to its lattice site. In their studies of this model Kenichiro Aoki and Dimitri Kusnezov emphasized its most interesting feature: because the quartic tethers act to scatter long-wavelength phonons, Φ4 chains exhibit Fourier heat conduction. In his recent Snook-Prize work Aoki also showed that the model can exhibit chaos on the three-dimensional energy surface describing a two-body two-spring chain. That surface can include at least two distinct chaotic seas. Aoki pointed out that the model typically exhibits different kinetic temperatures for the two bodies. Evidently few-body Φ4 problems merit more investigation. Accordingly, the 2018 Prizes honoring Ian Snook (1945-2013) will be awarded to the author(s) of the most interesting work analyzing and discussing few-body Φ4 models from the standpoints of dynamical systems theory and macroscopic thermodynamics, taking into account the model’s ability to maintain a steady-state kinetic temperature gradient as well as at least two coexisting chaotic seas in the presence of deterministic chaos.

Słowa kluczowe

EN

Φ4 model; chaos; Lyapunov exponents; algorithms

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2018
• Tom: Vol. 24, No. 2
• Strony: 159--163
• Opis fizyczny: Bibliogr. 13 poz., rys.

Twórcy

autor

Hoover W. G.

• Ruby Valley Research Institute Highway Contract 60, Box 601, Ruby Valley, Nevada 89833, USA

autor

Hoover C. G.

• Ruby Valley Research Institute Highway Contract 60, Box 601, Ruby Valley, Nevada 89833, USA

Bibliografia

• [1] K. Aoki, D. Kusnezov, Lyapunov Exponents and the Extensivity of Dimensional Loss for Systems in Thermal Gradients, Physical Review E 68, 056204 (2003).
• [2] K. Aoki, Symmetry, Chaos, and Temperature in the One-Dimensional Lattice Φ4 Theory, Computational Methods in Science and Technology 24 (2018) = ar_iv 1801.02865. His ar_iv version 1 contains the configurational part of our Figure 1. That figure is missing in version 2 and the internet version in CMST.
• [3] Wm.G. Hoover, C.G. Hoover, The 2017 SNOOK PRIZES in Computational Statistical Mechanics, Computational Methods in Science and Technology 24 (2018).
• [4] T. Hofmann, J. Merker, On Local Lyapunov Exponents of Chaotic Hamiltonian Systems, Computational Methods in Science and Technology 24 (2018).
• [5] G. Benettin, L. Galgani, A. Giorgilli, J.-M. Strelcyn, Lyapunov Characteristic Exponents for Smooth Dynamics Systems and for Hamiltonian Systems; a Method for Computing All of Them, Parts I and II: Theory and Numerical Application, Meccanica 15, 9–20 and 21–30 (1980).
• [6] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic Problems of Dissipative Dynamical Systems, Progress of Theoretical Physics 61, 1605–1616 (1979).
• [7] J.D. Farmer, E. Ott, J.A. Yorke, The Dimension of Chaotic Attractors, Physica D 7, 153–180 (1983).
• [8] Wm.G. Hoover, C.G. Hoover, Nonequilibrium Temperature and Thermometry in Heat-Conducting _4 Models, Physical Review E 77, 041104 (2008).
• [9] Wm.G. Hoover, C.G. Hoover, Microscopic and Macroscopic Simulation Techniques – Kharagpur Lectures, Section 10.8 (World Scientific Publishers, Singapore, 2018).
• [10] Wm.G. Hoover, K. Aoki, C.G. Hoover, S.V. De Groot, Time-Reversible Deterministic Thermostats, Physica D 187, 253–267 (2004).
• [11] M. Creutz, Microcanonical Monte Carlo Simulation, Physical Review Letters 50, 1411–1414 (1983).
• [12] Wm.G. Hoover, C.G. Hoover, Comparison of Very Smooth Cell-Model Trajectories Using Five Symplectic and Two Runge-Kutta Integrators, Computational Methods in Science and Technology 21 109–116 (2015). Note that the second-order Leapfrog algorithm at the top of page 111 is missing a factor of dt2 just after the “” sign.
• [13] Wm.G. Hoover, C.G. Hoover, Ergodic Isoenergetic Molecular Dynamics for Microcanonical-Ensemble Averages (ar_iv 1806.09802, 2018).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).