Perpetual motion of a mobile impurity in a one-dimensional quantum gas

TL;DR

This paper proves that an impurity in a one-dimensional quantum gas can exhibit perpetual motion, with its long-term momentum bounded below, under general conditions without approximations.

Contribution

It establishes a rigorous lower bound on the impurity's infinite-time momentum, demonstrating perpetual motion in a 1D quantum gas at zero temperature.

Findings

Lower bound on impurity momentum derived

Perpetual motion of impurity proven under general conditions

Bound simplifies for purely repulsive interactions

Abstract

Consider an impurity particle injected in a degenerate one-dimensional gas of noninteracting fermions (or, equivalently, Tonks-Girardeau bosons) with some initial momentum $p_0$. We examine the infinite-time value of the momentum of the impurity, $p_\infty$, as a function of $p_0$. A lower bound on $|p_\infty(p_0)|$ is derived under fairly general conditions. The derivation, based on the existence of the lower edge of the spectrum of the host gas, does not resort to any approximations. The existence of such bound implies the perpetual motion of an impurity in a one-dimensional gas of noninteracting fermions or Tonks-Girardeau bosons at zero temperature. The bound has an especially simple and useful form when the interaction between the impurity and host particles is everywhere repulsive.