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Atomistic simulations rely on statistical mechanical models. As such they may be performed within three main kinds of statistical ensembles. The canonical NPT ensemble (constant number of atoms, pressure, and temperature) is chiefly used in heating or quenching cycles. In contrast, the micro‐canonical NVE ensemble (constant number of atoms, volume, and energy) is primarily used when properties are calculated within the precise framework of statistical mechanics. As for the grand canonical μVT ensemble (constant chemical potential, volume, and temperature), it is typically used to investigate chemical equilibrium in systems that can exchange energy and matter with a reservoir.

For an isolated macroscopic system made up of a very large number N of atoms, each having three degrees of freedom, the microscopic state at a given instant is completely specified by the values of 3N coordinates r(i), collectively denoted by r ^N , and 3N momenta p(i), denoted similarly by p ^N . The values of the variables r ^N and p ^N define a point in a 6N‐dimensional space, called the phase space, symbolized by Γ_N. If H(r ^N , p ^N ) is the Hamiltonian of the system, the path followed by this point in the phase space is determined by Hamilton's equations:

(1)

(2)

where i = 1,…,N. In principle, 6N coupled equations subjected to 6N initial conditions should be solved to specify the values of all r(i) and p(i) at a given time.

The first difficulty encountered is that the timescale of microscopic processes is ultimately controlled by the 10⁻¹⁴–10⁻¹⁵ second period of atomic vibrations. In atomistic simulation, a time integration with similar discrete steps thus is required to describe accurately the time evolution of a system. Even with today's most powerful computers, this constraint restricts simulations to low‐viscosity conditions at which relaxation times (cf. Chapter 3.7) are lower than about 10⁻⁶ seconds to be consistent with the calculational time steps.

It follows that the glass transition cannot be investigated as a number of calculation steps of the order 10¹⁸ would be required to deal with a relaxation time of ~10³ seconds. Likewise, crystal nucleation and growth or liquid–liquid phase separation take place too slowly to be subjected to atomistic simulations. Besides, simulated melts are quenched at cooling rates at least six orders of magnitude faster than the highest rates of ~10⁶ K/s achievable practically. The fictive temperatures of the simulated glasses that then be up to 1000 K higher than those of real glasses, which one should keep in mind when making any kind of comparison between both kinds of materials.

Besides, a second difficulty stems from the fact that N is of the order of the Avogadro number (6.02*10²³) for any macroscopic system. Experience shows, however, that accurate results can be obtained for systems of only a few hundred or thousand atoms as long as structural units bigger than the system itself are not involved in the processes investigated. To avoid either creating surface or setting up a wall (Figure 1), periodic boundary conditions are usually imposed in numerical simulations. The cubic box is replicated throughout space to form an infinite lattice. As an atom moves in the original box, its periodic image in each of the neighboring boxes moves in exactly the same way. As an atom leaves the central box, one of its images will enter through the opposite face.

To simulate accurately a noncrystalline configuration in the cell, a larger number of atoms are nonetheless preferable. When a small cell of around several hundred atoms is studied, periodic boundary conditions, for instance, prevent structural units larger than the cell from forming. As already alluded to, any structural unit or spatial fluctuation larger than the wavelength which is greater than half of the cell size cannot be calculated appropriately because the repeatedly arranged fragments would be involved in the results. On the other hand, the maximum number of atoms which can be currently processed is of the order of 10⁵–10⁶. To check whether the number of atoms or the size of simulation cell is sufficient or not, the best way is to double the number of atoms and to maintain the same density through an adjustment of the cell size and then to make sure that there is little change in the calculated structural data.

Figure 1 Schematic representation of periodic boundary conditions.