The $\phi^4$ Model, Chaos, Thermodynamics, and the 2018 SNOOK Prizes in Computational Statistical Mechanics

Abstract

The one-dimensional $\phi^4$ 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, $\phi^4$ 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 the two-body two-spring chain. That surface can include {\it at least two} distinct chaotic seas. Aoki pointed out that the model typically exhibits different kinetic temperatures for the two bodies. Evidently few-body $\phi^4$ 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 $\phi^4$ 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.