Dynamics of semiflexible and rigid particles .1. The velocity distribution and the Smoluchowski equation

In this paper we present a derivation of Langevin equations of motion for long semiflexible particles and the Smoluchowski equation together with the velocity distribution for suspended rigid polymers in the long-time limit. The starting point is the induced force method of Mazur and Bedeaux [Physica A 76, 235 (1976)] and Landau-Lifshitz [Fluid Mechanics (Pergamon, Oxford, 1987)] fluctuating hydrodynamics. Such a procedure permits us to introduce in the description all the properties of the dynamics of the solvent in a rather straightforward way, which leads us to a precise derivation of friction coefficients, without assumptions taken out of the theory itself, and to a description in terms of Langevin equations. The link between the mesoscopic hydrodynamic description and a more coarse-grained one in terms of the Smoluchowski equation is thus established by means of a singular perturbation method. The long-time limit in the dynamics of the suspended particles permits us to also obtain the velocity distribution, which is not Maxwellian as postulated in classical treatments of Brownian motion. The velocity distribution obtained in this way relates the dynamics of suspensions to the dynamics of simple liquids. In addition, buoyancy and centrifugal forces are also obtained.