The variational conditions implied by the most probable equilibrium distribution for a dilute gas are set up exactly in terms of the digamma function without necessarily invoking a Stirling approximation. Through a sequence of lemmas it is proved that, at any given kinetic temperature, there are three classes of self-consistent solutions characterized by the parameterβ\(
\beta \bar \gtrless 0
\) 0 and by non-Maxwellian tails. These ambiguities persist even for a free ideal gas.