  To obtain further insight on possible power law generalizations of Boltzmann
equilibrium concepts, a stochastic collision model is investigated. We consider
the dynamics of a tracer particle of mass $M$, undergoing elastic collisions
with ideal gas particles of mass $m$, in the Rayleigh limit $m<<M$. The
probability density function (PDF) of the gas particle velocity is
$f(\tilde{v}_m)$. Assuming a uniform collision rate and molecular chaos, we
obtain the equilibrium distribution for the velocity of the tracer particle
$W_{eq}(V_M)$. Depending on asymptotic properties of $f(\tilde{v}_m)$ we find
that $W_{eq}(V_M)$ is either the Maxwell velocity distribution or a L\'evy
distribution. In particular our results yield a generalized Maxwell
distribution based on L\'evy statistics using two approaches. In the first a
thermodynamic argument is used, imposing on the dynamics the condition that
equilibrium properties of the heavy tracer particle be independent of the
coupling $\epsilon=m/M$ to the gas particles, similar to what is found for a
Brownian particle in a fluid. This approach leads to a generalized temperature
concept. In the second approach it is assumed that bath particles velocity PDF
scales with an energy scale, i.e. the (nearly) ordinary temperature, as found
in standard statistical mechanics. The two approaches yield different types of
L\'evy equilibrium which merge into a unique solution only for the
Maxwell--Boltzmann case. Thus, relation between thermodynamics and statistical
mechanics becomes non-trivial for the power law case. Finally, the relation of
the kinetic model to fractional Fokker--Planck equations is discussed.
