Figure 1

Example of reconstruction of a preassigned (analytic) free-energy profile (black curve), $F(E)$, by means of metadynamics runs consisting of 200 Gaussians of spread $\Delta E=0.4$ and height $w=0.16$. To mimic the uncertainty on the thermodynamic force encountered in stochastic simulations, a Gaussian noise of width 0.3 was added to $F^{\prime }(E)$. The broken red and indigo lines denote the filled profile, $\delta F(E)=F(E)-F_R(E)$, obtained, respectively, with an unsmoothed (${\tau }_c=0$) and smoothed (${\tau }_c=100$) metadynamics. Notice the shorter correlation lengths and smaller amplitudes of the fluctuations in the second case. The average and dispersion of $\delta F(E)$ were calculated (blue and brown lines wth error bars) for 1000 independent runs, revealing both the absence of biases in $F_R(E)$ and the constancy of the error except close to the boundaries. By changing $w$ and modifying the number of Gaussians so as to work at a constant filled free-energy volume, we calculated over the range $|E|<2.5$ the average, maximum, and minimum values of $\delta F(E)$ measured over 1000 runs; see inset. The linear dependence of the error on $F_R$ is thus apparent both for the unsmoothed and smoothed metadynamics (red and blue, respectively).Reuse & Permissions

Figure 2

Results from the metadynamics runs carried out on the $32\times 32$ Ising system. Left, top panel: the exact and reconstructed entropies. In both cases the entropy has been normalized so that $\sum _E\exp [S(E)]=1$. The abscissa indicates the energy per spin. Left, bottom panel: Difference between the true and reconstructed entropy. The dispersion expected a priori on the reconstructed entropy is shown with a red line. Right panel: the exact and reconstructed specific heat and average energy (inset) as a function of temperature.Reuse & Permissions