Highly degenerate diffusions for sampling molecular systems

This work is concerned with sampling and computation of rare events in molecular systems. In particular, we present new methods for sampling the canonical ensemble corresponding to the Boltzmann-Gibbs probability measure. We combine an equation for controlling the kinetic energy of the system with a random noise to derive a highly degenerate diffusion (i.e. a diffusion equation where diffusion happens only along one or few degrees of freedom of the system). Next the concept of hypoellipticity is used to show that the corresponding Fokker-Planck equation of the highly degenerate diffusion is well-posed, hence we prove that the solution of the highly degenerate diffusion is ergodic with respect to the Boltzmann-Gibbs measure. We find that the new method is more efficient for computation of dynamical averages such as autocorrelation functions than the commonly used Langevin dynamics, especially in systems with many degrees of freedom. Finally we study the computation of free energy using an adaptive method which is based on the adaptive biasing force technique