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

A kinetic 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 number of special cases and to an approximate solution with the explicitly specified precision in a general case. Previously we have reported that the impurity reaches a non-thermal steady state, characterized by an impurity momentum $p_\infty$ depending on its initial momentum $p_0$. In the present paper the detailed derivation of $p_\infty(p_0)$ 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 $\sqrt{F}$ for a heavy impurity.