Brownian dynamics with constraints

Systems possessing degrees of freedom operating on widely separated timescales, where the effects of those operating on the smaller timescales are relatively unimportant, may be modelled by the use of the Langevin equation. In order to study such systems containing complex polyatomic particles, holonomic constraints may be used. Though there is no lack of published algorithms for the numerical solution of the Langevin equation, few of them have been developed with sufficient rigour to ensure their precision, nor to demonstrate their compatibility with constraints. This study recapitulates an approach based upon Runge-Kutta equations which has the advantage of being perfectible to any desired order in the time-step, and shows how it may be combined with the SHAKE method in order to perform constrained Brownian dynamics simulations. Results are presented for some simple systems with a third order algorithm, and it is found that the correct dynamic and statistical behaviour is recovered.