  We use a set of cosmological simulations combined with radiative transfer
calculations to investigate the distribution of neutral hydrogen in the
post-reionization Universe. We assess the contributions from the metagalactic
ionizing background, collisional ionization and diffuse recombination radiation
to the total ionization rate at redshifts z=0-5. We find that the densities
above which hydrogen self-shielding becomes important are consistent with
analytic calculations and previous work. However, because of diffuse
recombination radiation, whose intensity peaks at the same density, the
transition between highly ionized and self-shielded regions is smoother than
what is usually assumed. We provide fitting functions to the simulated
photoionization rate as a function of density and show that post-processing
simulations with the fitted rates yields results that are in excellent
agreement with the original radiative transfer calculations. The predicted
neutral hydrogen column density distributions agree very well with the
observations. In particular, the simulations reproduce the remarkable lack of
evolution in the column density distribution of Lyman limit and weak damped
Ly\alpha\ systems below z = 3. The evolution of the low column density end is
affected by the increasing importance of collisional ionization with decreasing
redshift. On the other hand, the simulations predict the abundance of strong
damped Ly\alpha\ systems to broadly track the cosmic star formation rate
density.
