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We perform an extensive and detailed analysis of the generalized diffusionprocesses in deterministic area preserving maps with noncompact phase space,exemplified by the standard map, with the special emphasis on understanding theanomalous diffusion arising due to the accelerator modes. The accelerator modesand their immediate neighborhood undergo ballistic transport in phase space,and also the greater vicinity of them is still much affected ("dragged") bythem, giving rise to the non-Gaussian (accelerated) diffusion. The systematicapproach rests upon the following applications: the GALI method to detect theregular and chaotic regions and thus to describe in detail the structure of thephase space, the description of the momentum distribution in terms of theL\'evy stable distributions, the numerical calculation of the diffusionexponent and of the corresponding diffusion constant. We use this approach toanalyze in detail and systematically the standard map at all values of the kickparameter $K$, up to $K=70$. All complex features of the anomalous diffusionare well understood in terms of the role of the accelerator modes, mainly ofperiod 1 at large $K\ge 2\pi$, but also of higher periods (2,3,4,...) atsmaller values of $K\le 2\pi$.

Survey on the role of accelerator modes for anomalous diffusion: The case of the standard map