  Physical conditions within DLAs can reveal the star formation history,
determine the chemical composition of the associated ISM, and hence document
the first steps in the formation of present day galaxies. Here we present
calculations that self-consistently determine the gas ionization, level
populations (atomic fine-structure levels and rotational levels of H_2), grain
physics, and chemistry. We show that for a low-density gas (n< 0.1 cm^-3) the
meta-galactic UV background due to quasars is sufficient to maintain H_2 column
densities below the detection limit (i.e N(H_2)< 10^14 cm^-2) irrespective of
the metallicity and dust content in the gas. Such a gas will have a 21 cm spin
temperature in excess of 7000 K and very low C I and C II* column densities for
H I column densities typically observed in DLAs. We show that the observed
properties of the ~15% per cent of the DLAs that do show detectable H_2
absorption cannot be reproduced with only the quasar dominated meta-galactic UV
radiation field. Gas with higher densities (n>10 cm^-3), a moderate radiation
field (flux density one to ten times that of the background radiation of the
Galactic ISM), the observed range of metallicity and dust-to-gas ratio
reproduce all the observed properties of the DLAs that show H_2 absorption
lines. This favors the presence of ongoing star formation in DLAs with H_2. The
absence of detectable H_2 and C I absorption in a large fraction of DLAs can be
explained if they originate either in a low-density gas or in a high-density
gas with a large ambient radiation field. The absence of 21 cm absorption and C
II* absorption will be consistent with the first possibility. The presence of
21 cm absorption and strong C II* without H_2 and C I absorption will suggest
the second alternative. (Abridged)
