Molecular Dynamics Simulation

We saw that the Metropolis Monte Carlo simulation technique generates a sequence of states with appropriate probabilities for computing ensemble averages (Eq. 1). Generating states probabilitistically is not the only way to explore phase space. The idea behind the Molecular Dynamics (MD) technique is that we can observe our dynamical system explore phase space by solving all particle equations of motion. We treat the particles as classical objects that, at least at this stage of the course, obey Newtonian mechanics. Not only does this in principle provide us with a properly weighted sequence of states over which we can compute ensemble averages, it additionally gives us time-resolved information, something that Metropolis Monte Carlo cannot provide. The “ensemble averages” computed in traditional MD simulations are in practice time averages:

$$\left<G\right> = \bar{G} = \frac{1}{N_{\rm samp}\Delta t}\sum_{i=1}^{N_{\rm samp}} G\left[r\left(t\right)\right]$$ (108)

The ergodic hypothesis partially requires that the measurement time, $\tau_{\rm meas} = N_{\rm samp}\Delta t$, is greater than the longest relaxation time, $\tau_r$, in the system. The price we pay for this extra information is that we must at least access if not store particle velocities in addition to positions, and we must compute interparticle forces in addition to potential energy. We will introduce and explore MD in this section.