Projection of diffusions on submanifolds: Application to mean force computation

We consider the problem of sampling a Boltzmann-Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold ∑ of ℝⁿ implicitly defined by N constraints q₁(x) = ⋯ = q_N(x) = 0 (N < n). This problem arises, for example, in systems subject to hard constraints or in the context of free energy calculations. We prove that the constrained stochastic differential equations (i.e., diffusions) proposed in [7, 13] are ergodic with respect to this restricted distribution. We also construct numerical schemes for the integration of the constrained diffusions. Finally, we show how these schemes can be used to compute the gradient of the free energy associated with the constraints.