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We study an overdamped Langevin equation on the $$d$$-dimensional torus with stationary distribution proportional to $$p = e^{-U / \kappa}$$. When $$U$$ has multiple wells the mixing time of the associated process is exponentially large (of size $$e^{O(1/\kappa)}$$). We add a drift to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the mixing time. Our main result shows that the mixing time of the Langevin system can be made arbitrarily small by adding a drift that is sufficiently mixing. We provide one construction of a mixing drift, and our main result can be applied by using this drift with a large amplitude. For numerical purposes, it is useful to keep the size of the imposed drift small, and we show that the smallest allowable amplitude ensures that the mixing time is $$O( d/\kappa^2)$$, which is an order of magnitude smaller than $$e^{O(1/\kappa)}$$.
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