  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. We show that an exponentially mixing drift can be rescaled to make
the mixing time of the Langevin system arbitrarily small. For numerical
purposes, it is useful to keep the size of the imposed drift small, and we show
that the smallest allowable rescaling ensures that the mixing time is $O(
d/\kappa^2)$, which is an order of magnitude smaller than~$e^{O(1/\kappa)}$.
  We provide one construction of an exponentially mixing drift, although with
rate constants whose~$\kappa$-dependence is unknown.Heuristics (from discrete
time) suggest that $\kappa$-dependence of the mixing rate is such that the
imposed drift is of size~$O(d / \kappa^3)$. The large amplitude of the imposed
drift increases the numerical complexity, and thus we expect this method will
be most useful in the initial phase of Monte Carlo methods to rapidly explore
the state space.
