Use of Cartesian rotation vectors in Brownian dynamics algorithms: Theory and simulation results

We have shown that the components of Cartesian rotation vectors can be used successfully as generalized coordinates describing angular orientation in Brownian dynamics simulations of non-spherical nanoparticles. For this particular choice of generalized coordinates, we rigorously derived the conformation-space diffusion equations from kinetic theory for both free nanoparticles and nanoparticles interconnected by springs or holonomic constraints into polymer chains. The equivalent stochastic differential equations were used as a foundation for the Brownian dynamics algorithms. These new algorithms contain singularities only for points in the conformation-space where both the probability density and its first coordinate derivative equal zero (weak singularities). In addition, the coordinate values after a single Brownian dynamics time step are throughout the conformation-space equal to the old coordinate values plus the respective increments. For some parts of the conformation-spac these features represent a major improvement compared to the situation when Eulerian angles describe rotational dynamics. The presented simulation results of the equilibrium probability density for free nanoparticles are in perfect agreement with the results from kinetic theory.