  Cornerstone models of Physics, from the semi-classical mechanics in atomic
and molecular physics to planetary systems, are represented by quasi-integrable
Hamiltonian systems. Since Arnold's example, the long-term diffusion in
Hamiltonian systems with more than two degrees of freedom has been represented
as a slow diffusion within the `Arnold web', an intricate web formed by chaotic
trajectories. With modern computers it became possible to perform numerical
integrations which reveal this phenomenon for moderately small perturbations.
Here we provide a semi-analytic model which predicts the extremely slow-time
evolution of the action variables along the resonances of multiplicity one. We
base our model on two concepts: (i) By considering a (quasi-)stationary phase
approach to the analysis of the Nekhoroshev normal form, we demonstrate that
only a small fraction of the terms of the associated optimal remainder provide
meaningful contributions to the evolution of the action variables. (ii) We
provide rigorous analytical approximations to the Melnikov integrals of terms
with stationary or quasi-stationary phase. Applying our model to an example of
three degrees of freedom steep Hamiltonian provides the speed of Arnold
diffusion, as well as a precise representation of the evolution of the action
variables, in very good agreement (over several orders of magnitude) with the
numerically computed one.
