We study a free energy computation procedure, introduced in
[Darve and Pohorille,
J. Chem. Phys.115 (2001) 9169–9183; Hénin and Chipot,
J. Chem. Phys.121 (2004) 2904–2914], which relies on the long-time
behavior of a nonlinear stochastic
differential equation. This nonlinearity comes from a conditional
expectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions to
this equation has been proved
in [Lelièvre et al.,
Nonlinearity21 (2008) 1155–1181], under some existence and regularity assumptions.
In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear equation, and
we study a particle approximation technique based on a Nadaraya-Watson estimator of
the conditional expectation. The particle system converges to the solution
of the nonlinear equation if the number of particles goes to infinity
and then the kernel used in the Nadaraya-Watson approximation tends to a
Dirac mass.
We derive a rate for this convergence, and illustrate it by numerical
examples on a toy model.