  We derive an exact description of the non-equilibrium dynamics at finite
temperature for the anyonic Tonks-Girardeau gas extending the results of Atas
et al. [Phys. Rev. A 95, 043622 (2017)] to the case of arbitrary statistics.
The one-particle reduced density matrix is expressed as the Fredholm minor of
an integral operator with the kernel being the one-particle Green's function of
free fermions at finite temperature and the statistics parameter determining
the constant in front of the integral operator. We show that the numerical
evaluation of this representation using Nystr\"{o}m's method significantly
outperforms the other approaches present in the literature when there are no
analytical expressions for the overlaps of the wave-functions. We illustrate
the distinctive features and novel phenomena present in the dynamics of anyonic
systems in two experimentally relevant scenarios: the quantum Newton's cradle
setting and the breathing oscillations initiated by a sudden change of the trap
frequency.
