  We analyze the dynamics of a Brownian gas in contact with a heat bath in
which large temperature fluctuations occur. There are two distinct time scales
present, one describes the decay of the fluctuations in the temperature and the
other one is associated with the establishment of local equilibrium. Although
the gas has reached local equilibrium, there exist large fluctuations in an
intensive parameter (temperature) which break the thermodynamic equilibrium
with the heat bath. Thus the decay of the fluctuations in the intensive
parameter is larger than the characteristic time for the establishment of local
equilibrium. We show that the dynamics of such large and intensive fluctuations
may be described by adopting a nonequilibrium thermodynamics approach with an
adequate formulation of local equilibrium. A coarsening procedure is then used
to contract the space of mesoscopic variables needed to describe the dynamics
of the gas and the extensive character of the description is lost. This
procedure allows us to derive an effective Maxwell-Boltzmann factor (EMBF) for
the Brownian gas, as has been recently proposed in the literature. Furthermore,
we use this local equilibrium distribution and an entropy functional to derive
a nonequilibrium probability distribution and a hydrodynamic description for
the Brownian gas which contains fluctuating transport coefficients. The ensuing
description is nonextensive and our analysis shows that the coarse-graining
procedure is responsible for the nonextensivity property.
