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The scaling invariance for chaotic orbits near a transition from unlimited to limited 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 to suppress 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.
This is an easy-to-read introduction to foundations of deterministic chaos, deterministic diffusion and anomalous diffusion. The first part introduces to deterministic chaos in one-dimensional maps in form of Ljapunov exponents and dynamical entropie
Diffusion-limited aggregation is consistent with simple scaling. However, strong subdominant terms are present, and these can account for various earlier claims of anomalous scaling. We show this in detail for the case of multiscaling.
The Lorentz gas, a point particle making mirror-like reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent result
Lie group methods are applied to the time-dependent, monoenergetic neutron diffusion equation in materials with spatial and time dependence. To accomplish this objective, the underlying 2nd order partial differential equation (PDE) is recast as an ex
A scaling theory is developed for diffusion-limited cluster aggregation in a porous medium, where the primary particles and clusters stick irreversibly to the walls of the pore space as well as to each other. Three scaling regimes are predicted, conn