The Measurement of Free Energy by Monte Carlo Computer Simulation

Abstract

One of the most important problems in statistical mechanics is the measurement of free energies, these being the quantities that determine the direction of chemical reactions and--the concern of this thesis--the location of phase transitions. While Monte Carlo (MC) computer simulation is a well-established and invaluable aid in statistical mechanical calculations, it is well known that, in its most commonly-practised form (where samples are generated from the Boltzmann distribution), it fails if applied directly to the free energy problem. This failure occurs because the measurement of free energies requires a much more extensive exploration of the system's configuration space than do most statistical mechanical calculations: configurations which have a very low Boltzmann probability make a substantial contribution to the free energy, and the important regions of configuration space may be separated by potential barriers.

We begin the thesis with an introduction, and then give a review of the very substantial literature that the problem of the MC measurement of free energy has produced, explaining and classifying the various different approaches that have been adopted. We then proceed to present the results of our own investigations.

First, we investigate methods in which the configurations of the system are sampled from a distribution other than the Boltzmann distribution, concentrating in particular on a recently developed technique known as the multicanonical ensemble. The principal difficulty in using the multicanonical ensemble is the difficulty of constructing it: implicit in it is at least partial knowledge of the very free energy that we are trying to measure, and so to produce it requires an iterative process. Therefore we study this iterative process, using Bayesian inference to extend the usual method of MC data analysis, and introducing a new MC method in which inferences are made based not on the macrostates visited by the simulation but on the transitions made between them. We present a detailed comparison between the multicanonical ensemble and the traditional method of free energy measurement, thermodynamic integration, and use the former to make a high-accuracy investigation of the critical magnetisation distribution of the 2d Ising model from the scaling region all the way to saturation. We also make some comments on the possibility of going beyond the multicanonical ensemble to `optimal' MC sampling.

Second, we investigate an isostructural solid-solid phase transition in a system consisting of hard spheres with a square-well attractive potential. Recent work, which we have confirmed, suggests that this transition exists when the range of the attraction is very small (width of attractive potential/ hard core diameter ~ 0.01). First we study this system using a method of free energy measurement in which the square-well potential is smoothly transformed into that of the Einstein solid. This enables a direct comparison of a multicanonical-like method with thermodynamic integration. Then we perform extensive simulations using a different, purely multicanonical approach, which enables the direct connection of the two coexisting phases. It is found that the measurement of transition probabilities is again advantageous for the generation of the multicanonical ensemble, and can even be used to produce the final estimators.

Some of the work presented in this thesis has been published or accepted for publication: the references are

G. R. Smith & A. D. Bruce, A Study of the Multicanonical Monte Carlo Method, J. Phys. A. 28, 6623 (1995). [reference details doi:10.1088/0305-4470/28/23/015]

G. R. Smith & A. D. Bruce, Multicanonical Monte Carlo Study of a Structural Phase Transition, to be published in Europhys. Lett. [reference details Europhys. Lett. 34, 91 (1996) doi:10.1209/epl/i1996-00421-1]

G. R. Smith & A. D. Bruce, Multicanonical Monte Carlo Study of Solid-Solid Phase Coexistence in a Model Colloid, to be published in Phys. Rev. E [reference details Phys. Rev. E 53, 6530–6543 (1996) doi:10.1103/PhysRevE.53.6530]