  In this paper we propose a new formalism to map history-dependent
metadynamics in a Markovian process. We apply this formalism to a model
Langevin dynamics and determine the equilibrium distribution of a collection of
simulations. We demonstrate that the reconstructed free energy is an unbiased
estimate of the underlying free energy and analytically derive an expression
for the error. The present results can be applied to other history-dependent
stochastic processes such as Wang-Landau sampling.
