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

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