  The microphysical, kinetic properties of astrophysical plasmas near accreting
compact objects are still poorly understood. For instance, in modern
general-relativistic magnetohydrodynamic simulations, the relation between the
temperature of electrons $T_{e}$ and protons $T_{p}$ is prescribed in terms of
simplified phenomenological models where the electron temperature is related to
the proton temperature in terms of the ratio between the gas and magnetic
pressures, or $\beta$ parameter. We here present a very comprehensive campaign
of {two-dimensional} kinetic Particle-In-Cell (PIC) simulations of
special-relativistic turbulence to investigate systematically the microphysical
properties of the plasma in the trans-relativistic regime. Using a realistic
mass ratio between electrons and protons, we analyze how the index of the
electron energy distributions $\kappa$, the efficiency of nonthermal particle
production $\mathcal{E}$, and the temperature ratio $\mathcal{T}:=T_{e}/T_{p}$,
vary over a wide range of values of $\beta$ and $\sigma$. For each of these
quantities, we provide two-dimensional fitting functions that describe their
behaviour in the relevant space of parameters, thus connecting the
microphysical properties of the plasma, $\kappa$, $\mathcal{E}$, and
$\mathcal{T}$, with the macrophysical ones $\beta$ and $\sigma$. In this way,
our results can find application in wide range of astrophysical scenarios,
including the accretion and the jet emission onto supermassive black holes,
such as M87* and Sgr A*.
