Projection of diffusions on submanifolds: Application to mean force computation

Giovanni Ciccotti, Tony Lelievre, Eric Vanden-Eijnden

Research output: Contribution to journalArticlepeer-review


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) = ⋯ = qN(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.

Original languageEnglish (US)
Pages (from-to)371-408
Number of pages38
JournalCommunications on Pure and Applied Mathematics
Issue number3
StatePublished - Mar 2008

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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