Momentum space diffusion equations for chain molecules

A Fokker–Planck equation is derived for the reorientational and torsional motion of a flexible butane molecule. The analysis starts with the Liouville operator for a chain molecule in a fluid and by projection operator techniques yields an equation of motion for the reduced distribution function of the Euler angles, torsion angles, and the molecule-fixed angular and torsional momenta. A diffusion equation for butane is also derived by projection operator procedures. A fictitious metric potential arises in the diffusion equation and must be deleted by an offsetting potential. The time dependence of the diffusion coefficient, to lowest order in the streaming terms, can be obtained and yields a criteria for the applicability of Brownian motion models. The second moment of the torsion angle correlation function is also obtained and demonstrates a dependence on the dihedral potential, in contrast to the second moment of rigid body orientational correlation function which depends only on kBT and the appropriate components of the inertia tensor.