  We discuss inertial effects in systems outside equilibrium within the
framework of non-equilibrium thermodynamics. By introducing a Gibbs equation in
which the entropy depends on the probability density, we are able to describe a
system of Brownian particles immersed in a heat bath in both inertial and
diffusion regimes. In the former, a relaxation equation for the diffusion
current is obtained whereas in the latter we recover Fick's law. Our approach,
which uses the elements of the theory of internal degrees of freedom,
constitutes the mesoscopic version of a previous analysis which takes into
account the kinetic energy of diffusion.
