Surface chemistry in the Interstellar Medium II. $\mathrm{H}_2$
  formation on dust with random temperature fluctuations
  The $\mathrm{H}_2$ formation on grains is known to be sensitive to dust
temperature, which is also known to fluctuate for small grain sizes due to
photon absorption. We aim at exploring the consequences of simultaneous
fluctuations of the dust temperature and the adsorbed H-atom population on the
$\mathrm{H}_2$ formation rate under the full range of astrophysically relevant
UV intensities and gas conditions. The master equation approach is generalized
to coupled fluctuations in both the grain's temperature and its surface
population and solved numerically. The resolution can be simplified in the case
of the Eley-Rideal mechanism, allowing a fast computation. For the
Langmuir-Hinshelwood mechanism, it remains computationally expensive, and
accurate approximations are constructed. We find the Langmuir-Hinshelwood
mechanism to become an efficient formation mechanism in unshielded photon
dominated region (PDR) edge conditions when taking those fluctuations into
account, despite hot average dust temperatures. It reaches an importance
comparable to the Eley-Rideal mechanism. However, we show that a simpler rate
equation treatment gives qualitatively correct observable results in full cloud
simulations under most astrophysically relevant conditions. Typical differences
are a factor of 2-3 on the intensities of the $\mathrm{H}_2$ $v=0$ lines. We
also find that rare fluctuations in cloud cores are sufficient to significantly
reduce the formation efficiency. Our detailed analysis confirms that the usual
approximations used in numerical models are adequate when interpreting
observations, but a more sophisticated statistical analysis is required if one
is interested in the details of surface processes.
