  We investigate the large-time asymptotical behavior of a mobile impurity
immersed in a degenerate Tonks-Girardeau gas. We derive a correct weak-coupling
kinetic equation valid for arbitrary ratio of masses of gas and impurity
particles. When gas particles are either lighter or heavier than the impurity
we find that our theory is equivalent to the Boltzmann theory with the
collision integral calculated via the Fermi Golden Rule. On the contrary, in
the equal-mass case, Fermi Golden Rule treatment gives false results due to not
accounting for multiple coherent scattering events. The latter are treated by
the ressummation of ladder diagrams, which leads to a new kinetic equation. The
asymptotic momentum of the impurity produced from this equation coincides with
the result obtained by means of the Bethe ansatz.
