  It has recently been shown that turbulence in the interstellar medium (ISM)
can significantly accelerate the growth of dust grains by accretion of
molecules, but the turbulent gas-density distribution also plays a crucial role
in shaping the grain-size distribution. The growth velocity, i.e., the rate of
change of the mean grain radius, is proportional to the local gas density if
the growth species (molecules) are well-mixed in the gas. As a consequence,
grain growth happens at vastly different rates in different locations, since
the gas-density distribution of the ISM shows a considerable variance. Here, it
is shown that grain-size distribution (GSD) rapidly becomes a reflection of the
gas-density distribution, irrespective of the shape of the initial GSD. This
result is obtained by modelling ISM turbulence as a Markov process, which in
the special case of an Ornstein-Uhlenbeck process leads to a lognormal
gas-density distribution, consistent with numerical simulations of isothermal
compressible turbulence. This yields an approximately lognormal GSD; the sizes
of dust grains in cold ISM clouds may thus not follow the commonly adopted
power-law GSD with index -3.5, but corroborates the use of a log-nomral GSD for
large grains, suggested by several studies. It is also concluded that the very
wide range of gas densities obtained in the high Mach-number turbulence of
molecular clouds must allow formation of a tail of very large grains reaching
radii of several microns.
