Understanding the history and the evolution of the Milky Way disc is one of the main goals of modern astrophysics. We study the velocity dispersion behaviour of Galactic disc stars as a function of the [Mg/Fe] ratio, which can be used as a proxy of relative age. This key relation is essential to constrain the formation mechanisms of the disc stellar populations as well as the cooling processes. We used the recommended parameters and chemical abundances of 7800 FGK Milky Way field stars from the second internal data release of the Gaia-ESO Survey. These stars were observed with the GIRAFFE spectrograph, and cover a large spatial volume (6<R<10kpc and |Z|<2kpc). Based on the [Mg/Fe] and [Fe/H] ratios, we separated the thin- from the thick-disc sequence. From analysing the Galactocentric velocity of the stars for the thin disc, we find a weak positive correlation between Vphi and [Fe/H], due to a slowly rotating Fe-poor tail. For the thick disc, a strong correlation with [Fe/H] and [Mg/Fe] is established. We have detected an inversion of the radial velocity dispersion with [Mg/Fe] for thick-disc stars with [Fe/H]<-0.1dex and [Mg/Fe]>+0.2dex. First, the velocity dispersion increases with [Mg/Fe] at all [Fe/H] ratios for the thin-disc stars, and then it decreases for the thick-disc at the highest [Mg/Fe] abundances. Similar trends are observed within the errors for the azimuthal velocity dispersion, while a continuous increase with [Mg/Fe] is observed for the vertical velocity dispersion. The velocity dispersion decrease agrees with previous measurements of the RAVE survey, although it is observed here for a greater metallicity interval and a larger spatial volume. We confirm the existence of [Mg/Fe]-rich thick-disc stars with cool kinematics in the generally turbulent context of the primitive Galactic disc. This is discussed in the framework of the different disc formation scenarios.
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