  The departure of particle distributions from the Maxwellian is commonly
observed in space plasmas. These non-Maxwellian distributions which are typical
for plasmas that are not in thermal equilibrium, can be modeled with
$\kappa$-distribution. Kinetic simulations of quasi-parallel collisionless
shocks show that proton distribution is a composite of thermal, suprathermal,
and non-thermal parts. By using particle-in-cell shock simulations, we show
that $\kappa$-distribution adequately fits thermal and suprathermal parts
together, as a single continuous distribution in early proton spectra. We
derive suprathermal proton distribution directly from the generalized entropy
of non-extensive statistical mechanics, and show that thermal and suprathermal
populations are both naturally embedded in $\kappa$-distribution. We find that
the index $\kappa$ of the distribution increases with the distance from the
shock, following the decrease in suprathermal part. The non-equilibrium plasma
distribution which is continuously being enriched with suprathermal particles
at the reforming shock barrier, reaches the thermal equilibrium in the far
downstream. The suprathermal part completely fades there, and the shape of
proton distribution becomes a Maxwellian from which directly emerges a
power-law.
