  We investigate superdiffusion for stochastic processes generated by
nonuniformly hyperbolic system models, in terms of the convergence of rescaled
distributions to the normal distribution following the abnormal central limit
theorem, which differs from the usual requirement that the mean square
displacement grow asymptotically linearly in time. We construct a martingale
approximation that follows the idea of Doob's decomposition theorem. We obtain
an explicity formula for the superdiffusion constant in terms of the fine
structure that originates in the phase transitions as well as the geometry of
the configuration domains of the systems. Models that satisfy our main
assumptions include chaotic Lorentz gas, Bunimovich stadia, billiards with
cusps, and can be apply to other nonuniformly hyperbolic systems with slow
correlation decay rates of order $\cO(1/n)$.
