Mathematical Physics - Volume II - Numerical Methods

6.2 Molecular Dynamics

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which requires knowledge of the next position r i ( t + δ t ) and is susceptible to rounding error due to relatively large residue O ( δ t 2 ) . However, it is possible to obtain more accurate estimates of the velocity (and thus the kinetic energy of the system) using various modifications of the Verlet algorithm. The Störmer algorithm is a such modification of the Verlet algorithm, where the most pronounced computational advantage stems from the fact that at no time is the difference between two large numbers used to calculate a small number [5]. The computational scheme has the form r i ( t + δ t ) = r i ( t )+ δ t v i t + δ t 2 . (6.10) In addition to current positions and accelerations, recorded quantities include mid-step speeds v i t + δ t 2 = v i t − δ t 2 + δ t a i ( t ) . (6.11) Since the Störmer algorithm is only a modification of the Verlet algorithm, it produces identical trajectories. The problem, evident in expressions (6.10), arises because the velocities are not calculated at the same time points as their positions, which complicates the calculation of the total energy of the system. The above-mentioned algorithms, and similar ones available in literature, are com pletely adequate for most MD simulations. However, it is sometimes necessary to use higher-order integration schemes which use higher-order position vector derivatives in the Taylor approximation. These algorithms not only achieve higher calculation accuracy for the same time step but also allow use of a longer time step without losing accuracy (at least in a short run). Unfortunately, the use of these higher-order algorithms (an example is the popular predictor-corrector method) is coupled with many implementation difficulties that go beyond the scope of this introduction and have been discussed in detail in the literature (e.g., [5], [17]). Statistical physics provides the connection between microscopic behavior of the system and the macroscopic world described by thermodynamics. In order to calculate a certain physical parameter of the state of a macroscopic system (such as stress, strength, tempera ture, damage) it is necessary to define it as a function of the raw MD output data (that is, the atomic positions, velocity and forces ( r , v , a )). Strictly speaking, this can be achieved only when the thermodynamic system is: (i) large enough to be statistically homogeneous, and (ii) either in equilibrium or close enough to equilibrium (measured by the Deborah number [33]). If these preconditions are not met, the meaning of the continuum concepts becomes disputable.

6.2.4 Calculation of Macro-parameters of State