Generalized fluctuation-dissipation relation and statistics for the   equilibrium of a system with conformation dependent damping
  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. 
A. Bhattacharyay
cond-mat.stat-mech