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-Zeldovich (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
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