Metadynamics evolution

The continuous metadynamics algorithm can be applied to any system evolving under the action of a dynamics whose equilibrium distribution is canonical at an inverse temperature 1// . In a molecular dynamics scheme this requires that the evolution is carried out at constant temperature, by using a suitable thermostat [51]. In the continuous version of metad3mamics, Gaussians are added at every MD step and act directly on the microscopic variables. This generates at time t extra forces on x that can be written as... 

In order to understand how the algorithm actually works and to construct an explicit expression for the error it is not convenient to work with the metadynamics equations (12) in their full generality. Instead, we notice that the finite temperature dynamics of the collective variables satisfies, under rather general conditions, a stochastic differential equation [54,55]. Furthermore, in real systems the quantitative behavior of metadynamics is perfectly reproduced by the Langevin equation in its strong friction limit [56]. This is due to the fact that all the relaxation times are usually much smaller than the typical diffusion time in the CV space. Hence, we model the CVs evolution with a Langevin t3rpe dynamics ... 

As we checked in practical applications, the evolution of the collective variables can be modelled with a Langevin equation also for the Lagrangian metadynamics introduced in Sect. 2.1. Therefore, the error analysis performed in this section can be applied also for Lagrangian metadynamics. 

Figure 8 The time evolution of the chosen CV along the metadynamics (a) and the evolution of the bias (b).

Figure 12 Time evolution of the Ramachandran dihedral angle during metadynamics (a) and evolution of the deposited bias over time (b).

The parameters w and Gq set the strength and the width of the bias potential, respectively. By iterating the sequence of force calculations from a constrained MD simulation (or from a MC simulation) and the propagation step of (13) one obtains the time evolution of the collective variables. In the metadynamics terminology one then speaks of a walker traveling through the low-dimensional space of the collective variables on the FES F q). 

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