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Molecular dynamics

Molecular Dynamics method (MD) [42,54,55] has become a very powerful tool to attack many-body problems in statistical physics. This method allows studying specific aspects of complex systems in great detail via computer simulations. Simulations need specific input parameters that characterize the system in question, and which come either from theoretical models or from experimental data. These data help to fix the parameters of the model, the main part of which is interactions between atoms represented by an interatomic potential.

Once the potential is specified and the initial conditions, i.e. initial coordinates and velocities, are chosen the MD method can be applied. Molecular Dynamics are governed by the system's Hamiltonian $H$ and consequently by Hamiltonian's equations of motion:

\begin{displaymath}
\frac{dp_i}{dt}=-\frac{\partial H}{\partial q_i}
\end{displaymath} (4.1)


\begin{displaymath}
\frac{dq_i}{dt}=\frac{\partial H}{\partial p_i}
\end{displaymath} (4.2)

We integrate these equations to move particles to new positions $\{q_i\}$ and calculate new momenta $\{p_i\}$. The Hamiltonian of a system in classical MD simulations is usually written as a sum of the potential energy $V(\{q_i\})$, which depends on coordinates only, and the kinetic energy $K(\{p_i\})$, which is a function of momenta:
\begin{displaymath}
H=K(\{p_i\})+V(\{q_i\})=\sum_{i}\frac{p_i^2}{2m}+V({q_i})
\end{displaymath} (4.3)

The next step is the integration of these equations. For this purpose an integrator is needed, which propagates particle positions and velocities in time. It is a finite difference scheme, which moves particle trajectories discretely in time. The requirements for the integrator are the following: the integrator has to be accurate, in the sense that it approximates the true trajectory well enough, as well as it has to be stable, in the sense that it conserves the total energy and that small perturbations do not lead to instability. Different integrators could be used to solve Newton's equations differing in accuracy, complexity and speed. Among these the predictor-corrector ( PC ) [54-56], method was chosen for our simulations (See Appendix A).

Obviously, a statistical ensemble has to be chosen, where thermodynamic quantities like temperature, pressure and number of particles are controlled. There are algorithms, which fix temperature, volume or pressure to an appropriate value. In our simulations we use the canonical ensemble (NVT), with Nose-Hoover algorithm [60] which controls the temperature (See Appendix B) and the extended isothermal-isotension ensemble (NtT)(See Appendix C) of statistical mechanics [57, 58]. The last one is used to calculate the shape and the volume, as well as the shear elastic moduli(See Appendix D) of vanadium containing point defects.

The above mentioned steps essentially define the MD simulations. Having this tool at hand, it is possible in principle to obtain exact results, within numerical precision and round off computer errors. Of course, these results are only valid with respect to the model, which enters into simulations. The results of simulations have to be tested against theoretical predictions and experimental findings.

An important issue of MD simulations is the accessible time and length scale, which can be explored in MD simulations. It is clear that the more elaborate and detailed simulation technique is applied, the shorter is the time scale and smaller the length scale are accessible in simulations, due to limitation on the available CPU time and/or accessible computer memory. In classical molecular dynamics, in the case of solid state simulations, the time scale is dominated by the time scale of atomic vibrations (picosecond), and the accessible length scale is about $\simeq~100~\dot A$.

With the development of faster parallel architecture computers the accessible time and length scales are gradually increasing. J. Roth [43] demonstrated, for example, in 1999 on the Cray T3E-1200 computer in Julich, that it is possible to simulate more than 5,000,000,000 particles corresponding to the length scale of several thousand angstroms. In another demonstration run Y. Duan and A. Kollman [43] extended the time scale of a MD simulations up to 1 $\mu s$.


next up previous
Next: The interatomic potential Up: Numerical Methods Previous: Numerical Methods
2003-01-15