  We investigate the Joule expansion of an interacting quantum gas in an energy
eigenstate. The Joule expansion occurs when two subsystems of different
particle density are allowed to exchange particles. We demonstrate numerically
that the subsystems in their energy eigenstates evolves unitarily into the
global equilibrium state in accordance with the eigenstate thermalization
hypothesis. We find that the quantum gas changes its temperature after the
Joule expansion with a characteristic inversion temperature $T_{\rm I}$. The
gas cools down (heats up) when the initial temperature is higher (lower) than
$T_{\rm I}$, implying that $T_{\rm I}$ is a stable fixed point, which is
contrasted to the behavior of classical gases. Our work exemplifies that
transport phenomena can be studied at the level of energy eigenstates.
