  We present computer simulations about the spatial and temporal evolution of a
1-MeV proton microbeam transmitted through an insulating macrocapillary with
the length of 45 mm and with the inner diameter of 800 {\mu}m. The axis of the
capillary was tilted to 1{\deg} relative to the axis of the incident beam,
which ensured geometrical nontransparency. The simulation is based on the
combination of stochastic (Monte Carlo) and deterministic methods. It involves
(1) random sampling of the initial conditions, according to distributions
generated by the widely used and freely available computer software packages,
SRIM and WINTRAX, (2) the numerical solution of the governing equations for
following the classical trajectory of the projectiles, and (3) the description
of the field-driven charge migration on the surface and in the bulk of the
insulator material. We found that our simulation describes reasonably all of
our previous experimental observations, indicating the functionality and
reliability of the applied model. In addition, we found that at different
phases of the beam transmission, different atomic processes result in the
evolution of the beam distribution. First, in a scattering phase, the multiple
small angle atomic scattering dominates in the beam transmission, resulting in
an outgoing beam into a wide angular range and in a wide energy window. Later,
in a mixed phase, scattering and guiding happens simultaneously, with a
continuously increasing contribution of guiding. Finally, in the phase of the
stabilized, guided transmission, a quadrupolelike focusing effect is observed,
i.e., the transmitted beam is concentrated into a small spot, and the
transmitted protons keep their initial kinetic energy.
