Symmetry, Chaos and Temperature in the One-dimensional Lattice φ4 Theory

The symmetries of the minimal $\phi^4$ theory on the lattice, and trajectories which are chaotic, yet restricted to motions within subspaces due to symmetry reasons, are systematically analyzed. 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 are investigated. The pairing of local (finite-time) Lyapunov exponents are analyzed, and their dependence on various factors, such as energy of the system, characteristics of the initial conditions are studied.