Long-time convergence of an adaptive biasing force method: Variance reduction by Helmholtz projection

In this paper, we propose an improvement of the adaptive biasing force (ABF) method, by projecting the estimated mean force onto a gradient. We show on some numerical examples that the variance of the approximated mean force is reduced using this technique, which makes the algorithm more efficient than the standard ABF method. The associated stochastic process satisfies a nonlinear stochastic differential equation. Using entropy techniques, we prove exponential convergence to the stationary state of this stochastic process.