  Older GCE models predict [K/Fe] ratios as much as 1 dex lower than those
inferred from stellar observations. Abundances of potassium are mainly based on
analyses of the 7698 $\AA$ resonance line, and the discrepancy between models
and observations is in part caused by the LTE assumption. We study the
statistical equilibrium of KI, focusing on the non-LTE effects on the $7698 \
\AA$ line. We aim to determine how non-LTE abundances of K can improve the
analysis of its chemical evolution, and help to constrain the yields of models.
We construct a model atom that employs the most up-to-date data. In particular,
we calculate and present inelastic e+K collisional excitation cross-sections
from the convergent close-coupling and the $B$-Spline $R$-matrix methods, and
H+K collisions from the two-electron model. We constructed a fine grid of
non-LTE abundance corrections that span $4000<\teff / \rm{K}<8000$,
$0.50<\lgg<5.00$, $-5.00<\feh<+0.50$, and applied the corrections to abundances
from the literature. In concordance with previous studies, we find severe
non-LTE effects in the $7698 \ \AA$ line, which is stronger in non-LTE with
abundance corrections that can reach $\sim-0.7\,\dex$. We explore the effects
of atmospheric inhomogeneity by computing a full 3D non-LTE stellar spectrum of
KI for a test star. We find that 3D is necessary to predict a correct shape of
the resonance 7698 $\AA$ line, but the line strength is similar to that found
in 1D non-LTE. Our non-LTE abundance corrections reduce the scatter and change
the cosmic trends of literature K abundances. In the regime
[Fe/H]$\lesssim-1.0$ the non-LTE abundances show a good agreement with the GCE
model with yields from rotating massive stars. The reduced scatter of the
non-LTE corrected abundances of a sample of solar twins shows that line-by-line
differential analysis techniques cannot fully compensate for systematic
modelling errors.
