Role of nonextensive statistics in a three-degree-of-freedom gravitational system

We examine universal statistical patterns in the chaotic dynamics of three-degree-of-freedom Hamiltonians where long-range interactions are present. These statistical patterns are associated with the presence of a saddle-center-center critical point in the phase space of the systems, being basically discussed in the realm of closed inflationary cosmologies with two scalar fields. In the neighborhood of the critical point, the Hamiltonian is separable into two rotational motion sectors and one hyperbolic motion sector, with respective energies partially conserved. Because of the nonintegrability of the system, the distribution of these quantities for typical ensembles of orbits is chaotic, the associated distribution laws being in the realm of Tsallis nonextensive q thermostatistics. The nonintegrability parameters of the system, the masses m₁ and m₂ of the scalar fields, and the coupling parameter g² between the two scalar fields characterize the several routes of the system towards the high nonintegrability limit where the universal statistical patterns show up. A general feature is that the statistical partition of energy between the rotational sectors is made only via the gravitational sector, which plays the role of a reservoir of energy in the partition process. The dominance of the gravitational interaction (as ruled by the mass parameters) in the nonextensivity of the statistics over the interaction ruled by g² is established. As a striking example, we exhibit a route from Boltzmann-Gibbs behavior towards the nonextensive behavior as the gravitational interaction becomes dominant over the g² interaction.