Nicola Schneider, Volker Ossenkopf-Okada, Markus Roellig, Daniel Seifried, Ralf S. Klessen, Alexei G. Kritsuk, Eduard Keilmann, Simon Dannhauer, Lars Bonne, Simon C. O. Glover

In recent decades, significant attention has been dedicated to analytical and observational studies of the atomic hydrogen (HI) to molecular hydrogen (H2) transition in the interstellar medium. We focussed on the Draco diffuse cloud to gain deeper insights into the physical properties of the transition from HI to H2. We employed the total hydrogen column density probability distribution function (N-PDF) derived from Herschel dust observations and the N(HI)-PDF obtained from HI data collected by the Effelsberg HI survey. The N-PDF of the Draco cloud exhibits a double-log-normal distribution, whereas the N(HI)-PDF follows a single log-normal distribution. The HI-to-H2 transition is identified as the point where the two log-normal components of the dust N-PDF contribute equally; it occurs at Av = 0.33 (N=6.2e20 cm^-2). The low-column-density segment of the dust N-PDF corresponds to the cold neutral medium, which is characterized by a temperature of around 100 K. The higher-column-density part is predominantly associated with H2. The shape of the Draco N-PDF is qualitatively reproduced by numerical simulations. In the absence of substantial stellar feedback, such as radiation or stellar winds, turbulence exerts a significant influence on the thermal stability of the gas and can regulate the condensation of gas into denser regions and its subsequent evaporation. Recent observations of the ionized carbon line at 158 micron in Draco support this scenario. Using the KOSMA-tau photodissociation model, we estimate a gas density of n=50 cm^-3 and a temperature of 100 K at the location of the HI-to-H2 transition. Both the molecular and atomic gas components are characterized by supersonic turbulence and strong mixing, suggesting that simplified steady-state chemical models are not applicable under these conditions.