Order and Chaos in the One-Dimensional $\phi^4$ Model : N-Dependence and the Second Law of Thermodynamics

TL;DR

This paper explores the dynamical properties of the one-dimensional $4$ model, revealing stable periodic orbits, positive Lyapunov exponents, and discussing implications for the Second Law of Thermodynamics.

Contribution

It identifies stable periodic orbits with positive Lyapunov exponents and examines their role in understanding chaos and thermodynamics in the $4$ model.

Findings

Existence of infinite stable periodic orbits.

Some orbits exhibit positive Lyapunov exponents.

Perturbations can lead to chaotic behavior in extended phase space.

Abstract

We revisit the equilibrium one-dimensional $\phi^4$ model from the dynamical systems point of view. We find an infinite number of periodic orbits which are computationally stable. At the same time some of the orbits are found to exhibit positive Lyapunov exponents! The periodic orbits confine every particle in a periodic chain to trace out either the same or a mirror-image trajectory in its two-dimensional phase space. These "computationally stable" sets of pairs of single-particle orbits are either symmetric or antisymmetric to the very last computational bit. In such a periodic chain the odd-numbered and even-numbered particles' coordinates and momenta are either identical or differ only in sign. "Positive Lyapunov exponents" can and do result if an infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is introduced so that the motion expands into a four-dimensional phase space. In that extended space a positive exponent results. We formulate a standard initial condition for the investigation of the microcanonical chaotic number dependence of the model. We speculate on the uniqueness of the model's chaotic sea and on the connection of such collections of deterministic and time-reversible states to the Second Law of Thermodynamics.