3D chemical structure of diffuse turbulent ISM. I. Statistics of the
  HI-to-H$_2$ transition
  We studied the statistical properties of the HI-to-H$_2$ transition observed
in absorption in the local diffuse and multiphase ISM to identify the physical
processes controlling the probability of occurrence of any line of sight. The
turbulent diffuse ISM is modeled using the RAMSES code, which includes detailed
treatments of the magnetohydrodynamics, the thermal evolution of the gas, and
the chemistry of H$_2$. The impacts of the UV radiation field, the mean
density, the turbulent forcing, the integral scale, the magnetic field, and the
gravity on the molecular content of the gas are explored through a parametric
study covering a wide range of physical conditions. The statistics of the
HI-to-H$_2$ transition are interpreted through analytical prescriptions and
compared with the observations using a modified and robust version of the
Kolmogorov-Smirnov test. The results of one simulation, convolved with the
distribution of distances of the observational sample, are able to explain most
of the statistical properties of the HI-to-H$_2$ transition observed in the
local ISM. The tightest agreement is obtained for a neutral diffuse gas modeled
over ~200 pc, with a mean density of $1-2$ cm$^{-3}$, illuminated by the
standard interstellar UV radiation field, and stirred up by a large-scale
compressive turbulent forcing. Within this configuration, the 2D probability
histogram of the column densities of H and H$_2$ is remarkably stable and is
almost unaltered by gravity, the strength of the turbulent forcing, the
resolution of the simulation, or the strength of the magnetic field $B_x$. The
weak effect of the resolution and our analytical prescription suggest that the
column densities of HI are likely built up in large-scale WNM and CNM
structures correlated in density over ~20 pc and ~10 pc, respectively, while
those of H$_2$ are built up in CNM structures between ~3 pc and ~10 pc.
