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The computation of free energies is a common issue in statistical physics. Anatural technique to compute such high dimensional integrals is to resort toMonte Carlo simulations. However these techniques generally suffer from a highvariance in the low temperature regime, because the expectation is oftendominated by high values corresponding to rare system trajectories. A standardway to reduce the variance of the estimator is to modify the drift of thedynamics with a control enhancing the probability of rare event, leading toso-called importance sampling estimators. In theory, the optimal control leadsto a zero-variance estimator; it is however defined implicitly and computing itis of the same difficulty as the original problem. We propose here a generalstrategy to build approximate optimal controls in the small temperature limitfor diffusion processes, with the first goal to reduce the variance of freeenergy Monte Carlo estimators. Our construction builds upon low noiseasymptotics by expanding the optimal control around the instanton, which is thepath describing most likely fluctuations at low temperature. This technique notonly helps reducing variance, but it is also interesting as a theoretical toolsince it differs from usual small temperature expansions (WKB ansatz). As acomplementary consequence of our expansion, we provide a perturbative formulafor computing the free energy in the small temperature regime, which refinesthe now standard Freidlin--Wentzell asymptotics. We compute this expansionexplicitly for lower orders, and explain how our strategy can be extended to anarbitrary order of accuracy. We support our findings with illustrativenumerical examples.

Approximate optimal controls via instanton expansion for low temperature free energy computation