Monica Cerruti-Sola

Weak and strong chaos in Fermi-Pasta-Ulam models and beyond

We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions.

Hamiltonian dynamics and geometry of phase transitions in classical XY models

The Hamiltonian dynamics associated with classical, planar, Heisenberg XY models is investigated for two- and three-dimensional lattices. In addition to the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively different information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of Newtonian dynamics suggests consideration of other observables of geometric meaning tightly related to the largest Lyapunov exponent. The numerical computation of these observables--unusual in the study of phase transitions--sheds light on the microscopic dynamical counterpart of thermodynamics, also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces sigma E of phase space can be naturally established. In this framework, an approximate formula is worked out determining a highly nontrivial relationship between temperature and topology of sigma E. From this it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of sigma E. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.

On the clustering phase transition in self‐gravitating N‐body systems

The thermodynamic behaviour of self-gravitating N-body systems has been worked out by borrowing a standard method from molecular dynamics. The link between dynamics and thermodynamics is made in the microcanonical ensemble of statistical mechanics. Through the computation of basic thermodynamic observables and of the equation of state in the plane, the clustering phase transition appears to be of the second-order type. The dynamical–microcanonical averages are compared with their corresponding canonical ensemble averages, obtained through standard Monte Carlo computations. The latter seem to have completely lost any information about the phase transition. Finally, our results – obtained in a ‘microscopic’ framework – are compared with some existing theoretical predictions – obtained in a ‘macroscopic’ (thermodynamic) framework: qualitative and quantitative agreement is found, with an interesting exception.