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 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 $p_{\infty }$ 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 $p_{\infty }\left(p_0\right)$ 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, $m_{\mathrm{i}}>m_h$, damped oscillations of the impurity momentum develop, while in the opposite case, $m_{\mathrm{i}}<m_h$, 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.

Figure 1

Kinematically allowed regions for a single pairwise scattering in the cases of the light (left) and heavy (right) impurity. The final momentum of the impurity, $k$, is shown vs the initial momentum of the impurity, $q.$

Figure 2

The asymptotic impurity momentum $p_{\infty }$ as a function of the initial impurity momentum $p_0$ when impurity is lighter (left) and heavier (right) than a host particle. Solid blue line, an iterative solution (two iterations) of Eq. (31). This solution is exact below $q_2$ (not shown) and approximate above this point. The shaded area (green online) represents the maximal error: The exact solution of Eq. (31) lies inside this area. Notice much better convergence of iterations in the light impurity case.

Figure 3

Time evolution of the average momentum of the impurity moving under the action of a constant force. Initially the impurity is at rest. Solid curves, numerical solutions of the Boltzmann equation (44) with $f=0.05$; dashed lines, asymptotic momentum according to Eqs. (72) (light impurity) and (63) (heavy impurity). Oscillations are present in the heavy impurity case and absent in the light impurity case. The attenuation of the amplitude of the oscillations is shown by dotted line, see Eq. (50).

Figure 4

Average momentum of a steady state, $p_{\infty }$, vs dimensionless force $f$ applied to the impurity [see Eq. (43) for definition of $f$]. Black lines, asymptotical expressions in the small force limit according to Eqs. (72) and (63) for light and heavy impurity, correspondingly. Red dots, numerical solution of the steady-state Boltzmann equation (51). The steady-state momentum diverges at a critical force $f_{c1}$ (marked by a dashed vertical line), which is given by Eqs. (70) and (61) for light and heavy impurity, correspondingly.