The φ4 Model, Chaos, Thermodynamics, and the 2018 SNOOK Prizes in Computational Statistical Mechanics
Ruby Valley Research Institute
Highway Contract 60, Box 601, Ruby Valley, Nevada 89833, USA
E-mail: hooverwilliam@yahoo.com
Received:
Received: 09 June 2018; accepted: 11 June 2018; published online: 30 June 2018
DOI: 10.12921/cmst.2018.0000032
Abstract:
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.
Key words:
References:
[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, CMST 24(2), 83–95 (2018), arχiv 1801.02865. His arχiv 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(2), 55–79 (2018).
[4] T. Hofmann, J. Merker, On Local Lyapunov Exponents of Chaotic Hamiltonian Systems, CMST 24(2), 97–111 (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 φ4 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.
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.
Key words:
References:
[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, CMST 24(2), 83–95 (2018), arχiv 1801.02865. His arχiv 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(2), 55–79 (2018).
[4] T. Hofmann, J. Merker, On Local Lyapunov Exponents of Chaotic Hamiltonian Systems, CMST 24(2), 97–111 (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 φ4 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.
