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.
Chapter 2: Review
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 advantageous here for the generation of the multicanonical ensemble, and can even be used to produce the final estimators.
The thesis is available (1Mb file containing whole thesis (300 pages) as gzipped postscript, chapters not available separately).
J. Phys. A. Vol.28 No.23 pps.6623-6643 (1995).
We present a study of the multi-canonical Monte Carlo method which constructs and exploits Monte Carlo procedures that sample across an extended space of macrostates. We examine the strategies by which the sampling distribution can be constructed, showing in particular that a good approximation to this distribution may be generated efficiently by exploiting measurements of the transition rate between macrostates, in simulations launched from sub-dominant macrostates. We explore the utility of the method in the measurement of absolute free energies, and how it compares with traditional methods based on path-integration. We present new results revealing the behaviour of the magnetisation distribution of a critical finite-sized magnet, for magnetisation values extending from the the scaling region all the way to saturation.
PACS numbers: 05.70.Ce 02.70.Lq
Europhys. Lett. Vol.34 No.2 pps. 91-96 (1996).
We present a new method for establishing and exploiting a multi-canonical distribution for Monte Carlo sampling across an extended space of macrostates, with particular reference to the study of first-order phase transitions. The method exploits the information contained in measurements of the transition rate between macrostates, in simulations launched from macrostates with low canonical probabilities. We apply the method to the study of the coexistence of two solid phases in a model system of hard spheres with short-range attractive interactions.
PACS numbers: 02.70.Lq, 05.70.Ce
Phys. Rev. E. Vol.53 No.6 Pt.B pps. 6530-6543 (1996).
We describe a Monte Carlo approach to the determination of the relative stability of two phases, which is conceptually direct, potentially rather general, and particularly well suited to parallel computers. The approach exploits the information contained in the frequencies of the transitions between the macrostates of the order parameter distinguishing the two phases. The transition frequencies are observed in simulations initiated from macrostates with order-parameter-values intermediate between those of the two phases; they are used to provide estimators of the macrostate transition probability matrix, and thence estimators of the sampling distribution itself. The procedure allows one to construct a series of sampling distributions, weighted with respect to the canonical-distribution, which approach the multi-canonical limit, flat across order-parameter space. It entails only simulations that are short compared to the (multi-canonical) relaxation time of the order parameter. Reweighting the transition-probability estimator of the multi-canonical sampling distribution provides a good estimate of the canonical distribution of the order parameter for any value of the conjugate field, permitting the identification of the coexistence field in particular. The method is developed in the context of a system of hard-spheres with short range attractive interactions, described by a square potential well, which provides a simple model of the inter-colloid depletion potential in colloid-polymer mixtures. In particular we explore the phase diagram in the region in which studies by others, based on free energy evaluation by thermodynamic integration, have shown the coexistence of two fcc solid phases of different densities.
PACS numbers: 02.70.Lq, 05.70.Ce