  Liouville's theorem, based on the Hamiltonian flow (micro-canonical ensemble)
for a many particle system, indicates that the (stationary) equilibrium
probability distribution is a function of the Hamiltonian. A canonical ensemble
corresponds to a micro-canonical one at thermodynamic limit. On the contrary,
the dynamics of a single Brownian particle (BP) being explicitly
non-Hamiltonian with a force and damping term in it and at the other extreme to
thermodynamic limit admits the Maxwell-distribution (MD) for its velocity and
Boltmann-distribution (BD) for positions (when in a potential). This is due to
the fluctuation-dissipation relation (FDR), as was first introduced by
Einstein, which forces the Maxwell distribution to the Brownian particles. For
a structureless BP, that, this theory works is an experimentally verified fact
over a century now. Considering a structured Brownian particle we will show
that the BD and MD fails to ensure equilibrium. We will derive a generalized
FDR on the basis of the demand of zero current on inhomogeneous space. Our FDR
and resulting generalized equilibrium distributions recover the standard ones
at appropriate limits.

Statistical Mechanics