Metadynamics convergence law in a multidimensional system

Metadynamics is a powerful sampling technique that uses a nonequilibrium history-dependent process to reconstruct the free-energy surface as a function of the relevant collective variables s. In Bussi et al. [Phys. Rev. Lett. 96, 090601 (2006)] it is proved that, in a Langevin process, metadynamics provides an unbiased estimate of the free energy F(s). We here study the convergence properties of this approach in a multidimensional system, with a Hamiltonian depending on several variables. Specifically, we show that in a Monte Carlo metadynamics simulation of an Ising model the time average of the history-dependent potential converge to F(s) with the same law of an umbrella sampling performed in optimal conditions (i.e., with a bias exactly equal to the negative of the free energy). Remarkably, after a short transient, the error becomes approximately independent on the filling speed, showing that even in out-of-equilibrium conditions metadynamics allows recovering an accurate estimate of F(s). These results have been obtained introducing a functional form of the history-dependent potential that avoids the onset of systematic errors near the boundaries of the free-energy landscape.