Projection of diffusions on submanifolds: Application to mean force computation

Giovanni Ciccotti, Tony Lelievre, Eric Vanden-Eijnden

Research output: Contribution to journalArticle

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 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
Volume61
Issue number3
DOIs
StatePublished - Mar 2008

Fingerprint

Submanifolds
Free energy
Projection
Free Energy
Probability distributions
Gibbs Distribution
Differential equations
Sampling
Ludwig Boltzmann
Numerical Scheme
Stochastic Equations
Probability Distribution
Differential equation
Gradient
Context

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Communications on Pure and Applied Mathematics, Vol. 61, No. 3, 03.2008, p. 371-408.

Research output: Contribution to journalArticle

@article{947fd57fafda4095a06a620ebba45aff,
title = "Projection of diffusions on submanifolds: Application to mean force computation",
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 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.",
author = "Giovanni Ciccotti and Tony Lelievre and Eric Vanden-Eijnden",
year = "2008",
month = "3",
doi = "10.1002/cpa.20210",
language = "English (US)",
volume = "61",
pages = "371--408",
journal = "Communications on Pure and Applied Mathematics",
issn = "0010-3640",
publisher = "Wiley-Liss Inc.",
number = "3",

}

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 -