  Neglecting the effects associated with the solar wind plasma, the
photoelectrons are the only elementary particles which create an electrical
current through sunlit surface of the moon. They are knocked off of the surface
soil, rise above the surface, and then fall back. Therefore, on average, on any
unit of surface area there is a positive charge, equal in magnitude to the
charge of photoelectrons flying over this area. However, the charge of any
small dust particle can strongly fluctuate discretely: a photoelectron can be
either knocked off of the or be reacquired by the particle. The result is a
"random walk" in sign and magnitude of the charge of grains. In a few minutes
after sunrise, almost every dust particle on the surface has at least one extra
or missing electron, and the average modulus of the charge accumulated on a
particle is proportional to the square root of the number of "steps" (knocking
off /returns of photoelectrons). Therefore, the average value of the modulus of
the charge of a fine dust particle exceeds by several orders of magnitude the
proportion of the average surface charge attributable to the particle. So dust
particles that have ejected a sufficient number of photoelectrons can take off
from the surface because of the electric field of the near surface charge
double layer. It is shown that: (i) almost half of all dust particles on the
illuminated lunar surface are missing at least one electron; (ii) a significant
portion of particles up to 100 nm in size emits several photoelectrons and
acquire a positive charge, sufficient to take off from the surface; (iii) above
the surface there is a "boiling layer" of dust with a maximum thickness of
several hundred meters where the average size of the particles and their
density non-monotonically depend on the altitude.
