Monte Carlo calculations of the radial distribution functions for a proton-electron plasma

“In conclusion, the Monte Carlo method of calculating radial distribution functions in a plasma is a feasible approach if significant computing time is available (…) The results indicate that at least 10000 iterations must be completed before the system can be considered near to its equilibrium state, and for a badly chosen starting configuration, the run would need to be considerably longer (…) for more conclusive results a longer run is needed so that the energy of the system can settle into an equilibrium pattern and steady-state radial distribution functions can be obtained.” A.A. Barker

Looking for the history behind Barker’s formula the other day made me look for the original 1965 paper. Which got published in the Australian Journal of Physics at the beginning of Barker’s PhD at the University of Adelaide.

As shown in the above screenshot, the basis  of Barker’s algorithm is indeed Barker’s acceptance probability, albeit written in a somewhat confusing way since the current value of the chain is kept if a Uniform variate is smaller than what is actually the rejection probability. No mistake there! And more interestingly, Barker refers to Wood and Parker (1957) for the “complete and rigorous theory” behind the method. (Both Wood and Parker being affiliated with Los Alamos Scientific Laboratory, while Barker acknowledges support from both the Australian Institute of Nuclear Science and Engineering and the Weapons Research Establishment, Salisbury… This were times when nuclear weapon research was driving MCMC. Hopefully we will not come back to such times. Or, on the pessimistic side, we will not have time to come back to such times!)

As in Metropolis et al. (1953), the analysis is made on a discretised (finite) space, building the Markov transition matrix, stating the detailed balance equation (called microscopic reversibility). Interestingly, while Barker acknowledges that there are other ways of assigning the transition probability, his is the “most rapid” in terms of mixing. And equally interestingly, he discusses the scale of the random walk in the [not-yet-called] Metropolis-within-Gibbs move as major, targetting 0.5 as the right acceptance rate, and suggesting to adapt this scale on the go. There is also a side issue that is apparently not processed with all due rigour, namely the fact that the particles in the system cannot get arbitrarily close from one another. It is unclear how a proposal falling below this distance is processed by Barker’s algorithm. When implemented on 32 particles, this algorithm took five hours to execute 6100 iterations. With a plot of the target energy function that does not shout convergence, far from it! As acknowledged by Barker himself (p.131).

The above quote is from the conclusion and its acceptance of the need for increased computing times comes as a sharp contrast with this week when one of our papers was rejected based on this very feature..!