Topological aspects of geometrical signatures of phase transitions

Certain geometric properties of submanifolds of configuration space are numerically investigated for classical lattice phi^4 models in one and two dimensions. Peculiar behaviors of the computed geometric quantities are found only in the two-dimensional case, when a phase transition is present. The observed phenomenology strongly supports, though in an indirect way, a recently proposed topological conjecture about a topology change of the configuration space submanifolds as counterpart of a phase transition.Comment: REVTEX file, 4 pages, 5 figure

Analytical results for coupled map lattices with long-range interactions

We obtain exact analytical results for lattices of maps with couplings that decay with distance as $r^{-\alpha}$. We analyze the effect of the coupling range on the system dynamics through the Lyapunov spectrum. For lattices whose elements are piecewise linear maps, we get an algebraic expression for the Lyapunov spectrum. When the local dynamics is given by a nonlinear map, the Lyapunov spectrum for a completely synchronized state is analytically obtained. The critical lines characterizing the synchronization transition are determined from the expression for the largest transversal Lyapunov exponent. In particular, it is shown that in the thermodynamical limit, such transition is only possible for sufficiently long-range interactions, namely, for $\alpha\le alpha_c<d$, where $d$ is the lattice dimension.Comment: 4 pages, 2 figures, corrections included. Phys. Rev. E 68, 045202(R) (2003); correction in pres