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Projection of diffusions on submanifolds: Application to mean force computation
Author(s) -
Ciccotti Giovanni,
Lelièvre Tony,
VandenEijnden Eric
Publication year - 2008
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20210
Subject(s) - submanifold , mathematics , ergodic theory , context (archaeology) , stochastic differential equation , projection (relational algebra) , computation , distribution (mathematics) , mathematical analysis , algorithm , paleontology , biology
We consider the problem of sampling a Boltzmann‐Gibbs probability distribution when this distribution is restricted (in some suitable sense) on a submanifold Σ 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. © 2007 Wiley Periodicals, Inc.