  The Electron-Ion collider (EIC) will be the ultimate facility to study the
dynamics played by the colored quarks and gluons to the emergence of the global
phenomenology of the nucleons and nuclei as described by Quantum
Chromodynamics. The physics programs will greatly rely on efficient particle
identification (PID) in both the forward and the backward regions. The forward
and the backward RICHes of the EIC have to be able to cover wide acceptance and
momentum ranges; in the forward region a dual radiator RICH (dRICH) is foreseen
and in the backward region a proximity-focusing RICH can be foreseen to be
employed. The geometry and the performance studies of the dRICH have been
performed as prescribed in the EIC Yellow Report using the ATHENA software
framework. This part of our work reports the effort following the call for EIC
detector proposal the studies related to the forward and the backward RICHes
performance. In the forward region, dRICH performance showed a pion-kaon
separation from around 1 GeV/c to 50 GeV/c at a three sigma level; the
proximity focusing RICH (pfRICH) foreseen for the backward region can reach
three sigma separation up to 3 GeV/c for e/$\pi$ and up to 10 GeV/c for $\pi$/K
mass hypothesis.
