  The temperatures of interstellar dust grains are analyzed using stochastic
simulations, taking into account the grain composition and size and the
discreteness of the photon flux. [...] The distribution of grain temperatures
is calculated for a broad range of grain sizes and for different intensities of
the interstellar radiation field, relevant to diffuse clouds and to PDRs. The
dependence of the average grain temperature on its size is shown for different
irradiation intensities. It is found that the average temperatures of grains
with radii smaller than about 0.02 $\mu$m are reduced due to the fluctuations.
The average temperatures of grains of radii larger than about 0.35 $\mu$m are
also slightly reduced due to their more efficient emission of infrared
radiation, particularly when exposed to high irradiation intensities. The
average temperatures <T> of silicate and carbonaceous grains are found to
depend on the radiation field intensity X_MMP according to <T>~X_MMP^gamma,
where the exponent gamma depends on the grain size and composition. This
fitting formula is expected to be useful in simulations of interstellar
processes, and can replace commonly used approximations which do not account
for the grain temperature fluctuations and for the detailed properties of
interstellar dust particles. The implications of the results on molecular
hydrogen formation are also discussed. It is concluded that grain-temperature
fluctuations tend to reduce the formation rate of molecular hydrogen, and
cannot account for the observations of H_2 in photon dominated regions, even in
the presence of chemisorption sites.
