On the Stochastic Modeling of Rigid Body Systems with Application to Polymer Dynamics

The stochastic equations of motion for a system of interacting rigid bodies in a solvent are formulated and studied. Three-dimensional bodies of arbitrary shape, with arbitrary couplings between translational and rotational degrees of freedom, as arise in coarse-grained models of polymers, are considered. Beginning from an Euler–Langevin form of the equations, two different, properly invariant, Hamilton–Langevin forms are derived and studied together with various associated measures. Under different conditions depending on the choice of rotational coordinates, the canonical measure is shown to be a stationary solution of an associated Fokker–Planck equation and to always factorize into independent measures on configuration and velocity spaces. Explicit expressions are given for these measures, along with a certain Jacobian factor associated with the three-dimensional rotation group. When specialized to a fully coupled, quadratic model of a stiff polymer such as DNA, our results yield an explicit characterization of the complete set of model parameters.