  We probe the physical conditions in high-redshift damped Ly-alpha systems
using the observed molecular fraction and the rotational excitation of
molecular hydrogen. We report two new detections of H2 at z = 2.402 and 1.989
toward, respectively, HE 0027-1836 and HE 2318-1107. We also present a detailed
analysis of our recent H2 detection toward Q2343+125. All three systems have
low molecular fractions, log f < -4, with f = 2N(H2)/(2N(H2) + N(HI)). Only one
such H2 system was known previously. The depletion patterns for Si, S, Ti, Cr,
Mn, Fe and Ni in the three systems are found to be very similar to what is
observed in diffuse gas of the Galactic halo. H2 absorption from rotational
levels up to J = 5 is observed in a single component toward HE 0027-1836. We
show that the width (Doppler parameter) of the H2 lines increases with
increasing J and that the kinetic energy derived from the Doppler parameter is
linearly dependent on the relative energy of the rotational levels. The
excitation temperature is found to be 90 K for J = 0 to J = 2 and ~500 K for
higher J levels. Single isothermal PDR models fail to reproduce the observed
rotational excitations. A two-component model is needed: one component of low
density (~50 cm-3) with weak illumination (chi = 1) to explain the J <= 2
rotational levels and another of high density (~500 cm-3) with strong
illumination (chi = 30) for J >= 3 levels. However, the juxtaposition of these
two PDR components may be ad-hoc and the multicomponent structure could result
either from turbulent dissipation or C-shocks.
