On the distribution function of suprathermal particles at collisionless shocks

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.