Geometry of dynamics and phase transitions in classical latticeφ4theories

We perform a microcanonical study of classical lattice ${\phi }^4$ field models in three dimensions with O$\mathrm{}\left(n\right)\mathrm{}$ symmetries. The Hamiltonian flows associated with these systems that undergo a second-order phase transition in the thermodynamic limit are investigated here. The microscopic Hamiltonian dynamics neatly reveals the presence of a phase transition through the time averages of conventional thermodynamical observables. Moreover, peculiar behaviors of the largest Lyapounov exponents at the transition point are observed. A Riemannian geometrization of Hamiltonian dynamics is then used to introduce other relevant observables, which are measured as functions of both energy density and temperature. On the basis of a simple and abstract geometric model, we suggest that the apparently singular behavior of these geometric observables might probe a major topological change of the manifolds whose geodesics are the natural motions.