What is molecular dynamics?

  We call molecular dynamics (MD) a computer simulation technique where the time evolution of a set of interacting atoms is followed by integrating their equations of motion.

In molecular dynamics we follow the laws of classical mechanics, and most notably Newton's law:

$${\bf F}_i = m_i{\bf a}_i$$ (1)

for each atom i in a system constituted by N atoms. Here, m_i is the atom mass, ${\bf a}_i = d^2 {\bf r}_i/dt^2$its acceleration, and ${\bf F}_i$ the force acting upon it, due to the interactions with other atoms. Therefore, in contrast with the Monte Carlo method, molecular dynamics is a deterministic technique: given an initial set of positions and velocities, the subsequent time evolution is in principle completely determined. In more pictorial terms, atoms will ``move'' into the computer, bumping into each other, wandering around (if the system is fluid), oscillating in waves in concert with their neighbors, perhaps evaporating away from the system if there is a free surface, and so on, in a way pretty similar to what atoms in a real substance would do.

The computer calculates a trajectory in a 6N-dimensional phase space (3N positions and 3N momenta). However, such trajectory is usually not particularly relevant by itself. Molecular dynamics is a statistical mechanics method. Like Monte Carlo, it is a way to obtain a set of configurations distributed according to some statistical distribution function, or statistical ensemble. An example is the microcanonical ensemble, corresponding to a probability density in phase space where the total energy is a constant E:

$$\delta( H(\Gamma) - E ) .$$

Here, $H(\Gamma)$ is the Hamiltonian, and $\Gamma$ represents the set of positions and momenta. $\delta$ is the Dirac function, selecting out only those states which have a specific energy E. Another example is the canonical ensemble, where the temperature T is constant and the probability density is the Boltzmann function

$$\exp ( - H(\Gamma)/k_B T ) .$$

According to statistical physics, physical quantities are represented by averages over configurations distributed according to a certain statistical ensemble. A trajectory obtained by molecular dynamics provides such a set of configurations. Therefore, a measurements of a physical quantity by simulation is simply obtained as an arithmetic average of the various instantaneous values assumed by that quantity during the MD run.

Statistical physics is the link between the microscopic behavior and thermodynamics. In the limit of very long simulation times, one could expect the phase space to be fully sampled, and in that limit this averaging process would yield the thermodynamic properties. In practice, the runs are always of finite length, and one should exert caution to estimate when the sampling may be good (``system at equilibrium'') or not. In this way, MD simulations can be used to measure thermodynamic properties and therefore evaluate, say, the phase diagram of a specific material.

Beyond this ``traditional'' use, MD is nowadays also used for other purposes, such as studies of non-equilibrium processes, and as an efficient tool for optimization of structures overcoming local energy minima (simulated annealing).