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Quantum Boltzmann equation for a mobile impurity in a degenerate Tonks-Girardeau gas

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.