Autocorrelation Time and Equilibration Speed

Figure 9: Potential energy for Markov chains advanced with traditional Metropolis moves (red) and adding parallel tempering (black). Here, 8 replicas are run at temperatures ranging from 0.592 - 0.7 kcal/mol (300 - 375 K). The more rapid equilibration of the parallel tempering simulation is evident.

There are two ways in which parallel tempering can speed up a simulation. First, it may allow the system to equilibrate faster, as illustrated in Figure 9. Here, a single MC run with traditional Metropolis moves (at $300 K$) is started from the same initial configurations as a parallel tempering run with 8 replicas. The parallel tempering simulation reaches a stable equilibrium nearly $10$ times faster. Hence, the utility of parallel tempering in solving for the equilibrium sturctures of systems with frustrated geometries [4].

For an already equilibrated system, simulations can still be inefficient if configurations have a very long autocorrelation time. If parallel tempering can reduce the autocorrelation significantly, it could prove very useful for simulations of complex liquids like water. The estimated error in the mean from a set of data points is $\epsilon = \sqrt{\kappa/M}\sigma$, where $\sigma^2$ is the variance of the mean, $M$ is the number of sample points, and $\kappa$ is the autocorrelation time. If we have access to $P$ processors, we can run $P$ independent MC simulations of $M$ samples each and reduce the error estimate by $1/\sqrt{P}$. To beat this `` trivially parallel'' performance, a parallel tempering algorithm run on $P$ processors would have to reduce the autocorrelation time by a factor greater than $P$ to be useful. Our initial results (Figure 10) do not indicate such a performance boost, but they are inconclusive at this point.

Figure 10: Preliminary results for single - moleucle dipole - dipole autocorrelation times for simulations of 64 TIP5P water molecules using traditional Metroppolis MC (red) and parallel tempering (black, as described in Figure 9. Note: scales should be taken as arbitrary in this figure.