Symmetry, chaos and temperature in the one-dimensional lattice $\phi^4$ theory

TL;DR

This paper explores the chaotic dynamics, symmetries, and temperature behaviors in the one-dimensional lattice $$ theory, revealing how symmetries influence chaos, temperature distribution, and Lyapunov exponent pairing.

Contribution

It systematically analyzes the impact of symmetries on chaos and temperature in the lattice $$ theory, including novel insights into temperature differences and Lyapunov pairing.

Findings

Symmetries can restrict trajectories to subspaces while maintaining chaos.

Higher energies accelerate Lyapunov pairing times.

Symmetries can hinder the pairing and convergence of Lyapunov exponents.

Abstract

The symmetries of the minimal $\phi^4$ theory on the lattice are systematically analyzed. We find that symmetry can restrict trajectories to subspaces, while their motions are still chaotic. The chaotic dynamics of autonomous Hamiltonian systems are discussed, in relation to the thermodynamic laws. Possibilities of configurations with non-equal ideal gas temperatures in the steady state, in Hamiltonian systems, are investigated, and examples of small systems in which the ideal gas temperatures are different within the system are found. The pairing of local (finite-time) Lyapunov exponents are analyzed, and their dependence on various factors, such as the energy of the system, the characteristics of the initial conditions are studied, and discussed. We find that for the $\phi^4$ theory, higher energies lead to faster pairing times. We also find that symmetries can impede the pairing of local Lyapunov exponents, and the convergence of Lyapunov exponents.