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-timebehavior of a nonlinear stochasticdifferential equation. This nonlinearity comes from a conditionalexpectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions tothis equation has been provedin [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, andwe study a particle approximation technique based on a Nadaraya-Watson estimator ofthe conditional expectation. The particle system converges to the solutionof the nonlinear equation if the number of particles goes to infinityand then the kernel used in the Nadaraya-Watson approximation tends to aDirac mass. We derive a rate for this convergence, and illustrate it by numericalexamples on a toy model.