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

Generalized fluctuation-dissipation relation and statistics for the equilibrium of a system with conformation dependent damping