  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.
