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

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.