Application of the diffusion equation to prove scaling invariance on the transition from limited to unlimited diffusion

The scaling invariance for chaotic orbits near a transition from limited to unlimited diffusion in a dissipative standard mapping is explained via the analytical solution of the diffusion equation. It gives the probability of observing a particle with a specific action at a given time. We show the diffusion coefficient varies slowly with the time and is responsible for suppressing the unlimited diffusion. The momenta of the probability are determined and the behavior of the average squared action is obtained. The limits of small and large time recover the results known in the literature from the phenomenological approach and, as a bonus, a scaling for intermediate time is obtained as dependent on the initial action. The formalism presented is robust enough and can be applied in a variety of other systems including time-dependent billiards near a transition from limited to unlimited Fermi acceleration as we show at the end of the letter and in many other systems under the presence of dissipation as well as near a transition from integrability to non-integrability.