  We propose a new method to determine the electron velocity (EV) distribution
function in the intracluster gas (ICG) in clusters of galaxies based on the
frequency dependence of the Sunyaev-Zel'dovich (SZ) effect. It is generally
accepted that the relativistic equilibrium EV distribution is the one suggested
by Juttner. However, there is an ongoing debate on the foundation of
relativistic kinetic theory, and other distributions have also been proposed.
The mildly relativistic intracluster gas (ICG) provides a unique laboratory to
test relativistic kinetic theories. We carried out Monte Carlo simulations to
generate SZ signal from a single-temperature gas assuming the Juttner EV
distribution assuming a few per cent errors. We fitted SZ models based on
non-relativistic Maxwellian, and its two relativistic generalizations, the
Juttner and modified Juttner distributions. We found that a 1% error in the SZ
signal is sufficient to distinguish between these distributions with high
significance based on their different best-fit temperatures. However, in any
LOS in a cluster, the ICG contains a range of temperatures. Using our
N-body/hydrodynamical simulation of a merging galaxy cluster and assuming a 1%
error in the SZ measurements in a LOS through a bow shock, we find that it is
possible to distinguish between Juttner and modified Juttner distributions with
high significance. Our results suggest that deriving ICG temperatures from
fitting to SZ data assuming different EV distribution functions and comparing
them to the temperature in the same cluster obtained using other observations
would enable us to distinguish between the different distributions.11 pages, 8
figures, and 1 table, accepted for publication in the Astrophysical Journal
