  aims: We aim to provide an estimate of the minimum temperature of the quiet
solar chromosphere. methods: We perform a 2D radiation-MHD simulation spanning
the upper convection zone to the lower corona. The simulation includes non-LTE
radiative transfer and {an equation-of-state that includes non-equilibrium
ionization of hydrogen and non-equilibrium H_2 molecule formation}. We analyze
the reliability of the various assumptions made in our model in order to assess
the realism of the simulation. results: Our simulation contains pockets of cool
gas with down to 1660 K from 1 Mm up to 3.2 Mm height. It overestimates the
radiative heating, and contains non-physical heating below 1660 K. Therefore we
conclude that cool pockets in the quiet solar chromosphere might have even
lower temperatures than in the simulation, provided that there exist areas in
the chromosphere without significant magnetic heating. We suggest off-limb
molecular spectroscopy to look for such cool pockets and 3D simulations
including a local dynamo and a magnetic carpet to investigate Joule heating in
the quiet chromosphere.
