Kinetic theory for a mobile impurity in a degenerate Tonks-Girardeau gas

Akinetic theory describing the motion of an impurity particle in a degenerate Tonks-Girardeau gas is presented. The theory is based on the one-dimensional Boltzmann equation. An iterative procedure for solving this equation is proposed, leading to the exact solution in a number of special cases and to an approximate solution with the explicitly specified precision in a general case. Previously we reported that the impurity reaches a nonthermal steady state, characterized by an impurity momentum p(infinity) depending on its initial momentum p(0) [E. Burovski, V. Cheianov, O. Gamayun, and O. Lychkovskiy, Phys. Rev. A 89, 041601(R) (2014)]. In the present paper the detailed derivation of p(infinity)(p(0)) is provided. We also study the motion of an impurity under the action of a constant force F. It is demonstrated that if the impurity is heavier than the host particles, m(i) > m(h), damped oscillations of the impurity momentum develop, while in the opposite case, m(i) < m(h), oscillations are absent. The steady-state momentum as a function of the applied force is determined. In the limit of weak force it is found to be force independent for a light impurity and proportional to root F for a heavy impurity.