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An exact mathematical discussion is given of the spectrum of the operator governing the time evolution of the uniform rarefied rigid-sphere gas coupled to a thermal bath. Adapted methods, developed in the neutron thermalization theory and rarefied-gas dynamics, are used. Two cases are discussed in some detail: the thermal bath consists of gas particles of mass much smaller than the particles of the considered rarefied gas (i.e. the Brownian motion of the rarefied gas) and the thermal bath consists of gas particles of the same mass as the particles of the rarefied gas. The results are interpreted physically in connection with the possibility of a macroscopic description of the system.

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