  The inelastic Lorentz gas in cooling states is studied. It is found that the
inelastic Lorentz gas is localized and that the mean square displacement of the
inelastic Lorentz gas obeys a power of a logarithmic function of time. It is
also found that the scaled position distribution of the inelastic Lorentz gas
has an exponential tail, while the distribution is close to the Gaussian near
the peak. Using a random walk model, we derive an analytical expression of the
mean square displacement as a function of time and the restitution coefficient,
which well agrees with the data of our simulation. The exponential tail of the
scaled position distribution function is also obtained by the method of
steepest descent.
