Comparison of time- and length-scales, accessible to different types of simulation techniques (quantum simulations (QM), molecular dynamics (MD), Brownian Dynamics (BD) and hydrodynamics/fluid dynamics (HD))

Computing every force on each atom is a demanding part of a MD simulation. In order to limit the number of force calculations, the smallest possible system – in terms of number of atoms – should be simulated. The minimum system size necessary to compute a given desired property is not always obvious and a common practice is to analyze how increasing system size influences obtained results. A number of algorithmic approaches exist that help minimize system size such as applying periodic boundary conditions.
In addition to how large a system must be, one must determine how long a simulation must be run to obtain reliable answers (i.e. with sufficient accuracy and precision). Related to this is that the MD simulation time step should be small compared to the period associated with the highest frequency atomic event. A typical time step for MD simulations of metals is of order 1 fs (i.e. 1 E-15 s). The very small time step is the main drawback of MD method compared to other commonly used numerical simulations. Processes with time scale of order microseconds and larger are outside the reasonable realm for even highly parallel, efficient MD codes. Conversely, MC techniques cannot probe dynamics but they can probe long time equilibrium properties.

Figure 2.1 Schematic comparison of time- and length-scales, accessible to different types of simulation techniques (quantum simulations (QM), molecular dynamics (MD), Brownian Dynamics (BD) and hydrodynamics/fluid dynamics (HD))

Accessible length- and time- scales of microscopic simulation should be considered. Figure 2.1 shows a schematic representation for different types of simulations in a length-time-diagram. It is clear that the more detailed a simulation technique addresses, the smaller is the accessibility of times and length scales. Hence, quantum simulations (QS), where fast motion of electrons are considered, are located in the lower left corner of the diagram and typical length time scales are of order of Å (Å) and ps. MD method puts fixed partial charges on interaction sites or adds a dynamics model for polarization effects to simulate electronic distribution.  Hence, the time scale of the system is determined by the time of collision between atoms and atomic vibration frequencies. Consequently, the accessible time scale is of order ns and corresponding accessible length scale becomes of order 10-10000 Å. Note Figure 2.1 was originally presented in 2002; accessible length scales for MD have increased dramatically. Brownian Dynamics (BD) can be applied to deal with particles in a solvent medium, where one is not interested in a detailed description of the solvent. In BD the effect of the solvent is hidden in average quantities. If one is not interested in a molecular picture but in macroscopic quantities, the concepts of hydrodynamics (HD) may be applied, where the system properties are hidden in effective numbers, like density and sound velocity.