A symmetry break in energy distribution and a biased random walk behavior causing unlimited diffusion in a two dimensional mapping

We have shown that a break of symmetry of the probability distribution and a biased random walk behavior lead the dynamics of a two dimensional mapping to present unlimited diffusion. The mapping considered describes the dynamics of the Fermi-Ulam model forced by a stochastic perturbation. For the conservative dynamics and considering a high initial velocity/energy, we explain the changeover from a plateau to a regime of unlimited diffusion using arguments based on a break of symmetry of the probability distribution of the velocity and a biased random walk behavior for the energy. For the dissipative case we end up with a scaling result using arguments of steady state so far obtained numerically in the literature before. The break of symmetry used here can be extended to many other different models to explain transitions from limited to unlimited growth, including billiard problems.