### Abstract

We consider the problem of sampling a Boltzmann-Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold ∑ of ℝ^{n} implicitly defined by N constraints q_{1}(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.

Original language | English (US) |
---|---|

Pages (from-to) | 371-408 |

Number of pages | 38 |

Journal | Communications on Pure and Applied Mathematics |

Volume | 61 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2008 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Communications on Pure and Applied Mathematics*,

*61*(3), 371-408. https://doi.org/10.1002/cpa.20210

**Projection of diffusions on submanifolds : Application to mean force computation.** / Ciccotti, Giovanni; Lelievre, Tony; Vanden-Eijnden, Eric.

Research output: Contribution to journal › Article

*Communications on Pure and Applied Mathematics*, vol. 61, no. 3, pp. 371-408. https://doi.org/10.1002/cpa.20210

}

TY - JOUR

T1 - Projection of diffusions on submanifolds

T2 - Application to mean force computation

AU - Ciccotti, Giovanni

AU - Lelievre, Tony

AU - Vanden-Eijnden, Eric

PY - 2008/3

Y1 - 2008/3

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=38949150015&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=38949150015&partnerID=8YFLogxK

U2 - 10.1002/cpa.20210

DO - 10.1002/cpa.20210

M3 - Article

AN - SCOPUS:38949150015

VL - 61

SP - 371

EP - 408

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

SN - 0010-3640

IS - 3

ER -