Elementary Monte Carlo updates

The efficiency of Monte Carlo updates in continuous polymer and protein models strongly depends on the model chosen and can hardly be generalized. For this reason, we will describe only a few of the most basic and popular variants. 

The simplest one is the displacement update, in which the coordinates of a monomer at position i in the chain, x are modified by a random shift Ax,. The original conformation of a polymer with chain length N, X = xi, X2. x . xat, is changed by the update toX = xi, X2. X, -b Ax. xat. In Cartesian coordinates, the shift is performed by choosing uniformly distributed random numbers in the intervals Ax, e —dx, +dx, A , e [—dy,+dy, and Az, 6 —dz,+dz (with dx,dy,dz 0). For simplicity, it is common to use a cubic box with boundaries dx = dy = dz- Whether the update is accepted or rejected depends on the definition of the transition probability of the Monte Carlo method used. The displacement update is tried for all monomers in the chain (randomly or sequentially). An entire chain update is considered as a Monte Carlo sweep. 

The displacement update in Cartesian coordinates is apparently ergodic (each conformation can be reached out of any other) and satisfies the detailed balance condition [the ratio of selection probabilities (4,82) for updates X — X and X — X is unity]. The first condition is obeyed, because the integral over the volume element dV = j dxidyidzi yields 

It is very important to understand that the application of updates based on coordinates whose volume elements contain a non-uniform integral measure requires some care to avoid a violation of the detailed balance condition. The most prominent example 

An update that does not satisfy this condition is not ergodic and cannot be used as a single update. However, a combination of individually non-eigodic updates that as a whole cover the conformation space completely, is ergodic. 

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Statistical mechanical Monte Carlo as well as classical molecular dynamic methods can be used to simulate structure, sorption, and, in some cases, even diffusion in heterogeneous systems. Kinetic Monte Carlo simulation is characteristically different in that the simulations follow elementary kinetic surface processes which include adsorption, desorption, surface diffusion, and reactivity . The elementary rate constants for each of the elementary steps can be calculated from ab initio methods. Simulations then proceed event by event. The surface structure as well as the time are updated after each event. As such, the simulations map out the temporal changes in the atomic structure that occur over time or with respect to processing conditions. 

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