  Simplified assumptions about the thermodynamics of the electrons are normally
employed in general-relativistic magnetohydrodynamic (GRMHD) simulations of
accretion onto black holes. To counter this, we have developed a
self-consistent approach to study magnetised and radiatively cooled
two-temperature accretion flows around a Kerr black hole in two spatial
dimensions. The approach includes several heating processes, radiative cooling,
and a coupling between the electrons and the ions via Coulomb interaction. We
test our approach by performing axisymmetric GRMHD simulations of magnetised
tori accreting onto a Kerr black hole under various astrophysical scenarios. In
this way, we find that the inclusion of the Coulomb interaction and the
radiative cooling impacts the thermodynamical properties of both the ions and
electrons, changing significantly the temperature distribution of the latter,
and underlining the importance of a two-temperature approach when imaging these
flows. In addition, we find that the accretion rate influences the bulk
properties of the flow as well as the thermodynamics of the electrons and ions.
Interestingly, we observe qualitatively distinct temperature properties for
SANE and MAD accretion modes while maintaining the same accretion rates, which
could help distinguishing MAD and SANE accretion flows via observations.
Finally, we propose two new relations for the temperature ratios of the
electrons, ions, and of the gas in terms of the plasma-$\beta$ parameter. The
new relations represent a simple and effective approach to treat
two-temperature accretion flows on supermassive black holes such as Sgr A* and
M\,87*.
