The Measurement of Free Energy by MonteCarlo Computer Simulation Graham R Smith A thesis submitted in fullment of the requirements for the degree of Do ctor of Philosophy to the University of Edinburgh Abstract One of the most imp ortant problems in statistical mechanics is the measurement of free energies these b eing the quantities that determine the direction of chemical reactions andthe concern of this thesisthe lo cation of phase transitions While Monte Carlo MC computer simulation is a wellestablished and invaluable aid in statistical mechanical calculations it is well known that in its most commonlypractised form where samples are generated from the Boltzmann distribution it fails if applied directly to the free energy problem This failure o ccurs b ecause the measurement of free energies requires a much more extensive exploration of the systems conguration space than do most statistical mechanical calculations congurations which have a very low Boltzmann probability make a substantial contribution to the free energy and the imp ortant regions of conguration space may b e separated by p otential barriers We b egin 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 pro duced explaining and classifying the various dierent approaches that have b een adopted We then pro ceed to present the results of our own investigations First we investigate metho ds in which the congurations 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 diculty in using the multicanonical ensemble is the diculty 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 pro duce it requires an iterative pro cess Therefore we study this iterative pro cess using Bayesian inference to extend the usual metho d of MC data analysis and introducing a new MC metho d in which inferences are made based not on the macrostates visited by the simulation but on the transitions made b etween them We present a detailed comparison b etween the multicanonical ensemble and the traditional metho d of free energy measurement thermo dynamic integration and use the i former to make a highaccuracy investigation of the critical magnetisation distribution of the d Ising mo del from the scaling region all the way to saturation We also make some comments on the p ossibility of going b eyond the multicanonical ensemble to optimal MC sampling Second we investigate an isostructural solidsolid phase transition in a system consisting of hard spheres with a squarewell attractive p otential Recent work which we have conrmed suggests that this transition exists when the range of the attraction is very small width of attractive p otential hard core diameter First we study this system using a metho d of free energy measurement in which the squarewell p otential is smo othly transformed into that of the Einstein solid This enables a direct comparison of a multicanonicallike metho d with thermo dynamic integration Then we p erform extensive simulations using a dierent purely multicanonical approach which enables the direct connection of the two co existing phases It is found that the measurement of transition probabilities is again advantageous for the generation of the multicanonical ensemble and can even b e used to pro duce the nal estimators Some of the work presented in this thesis has b een 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 G R Smith A D Bruce Multicanonical Monte Carlo Study of a Structural Phase Tran sition to b e published in Europhys Lett G R Smith A D Bruce Multicanonical Monte Carlo Study of SolidSolid Phase Coexis tence in a Model Col loid to b e published in Phys Rev E ii Declaration This thesis has b een comp osed by myself and it has not b een submitted in any previous ap plication for a degree The work rep orted within was executed by me unless otherwise stated March iii for Christina and Ken iv Acknowledgements I would like to thank the following p eople Alastair Bruce for all his guidance help and en couragement and for never shouting at me even when I richly deserved it Stuart Pawley and Nigel Wilding for many useful and pleasant discussions David Greig Stuart Johnson Stephen Bond and Stephen Ilett for carefully reading and commenting on the nal draft of this thesis Peter Bolhuis for making available the results of my atmates and all my other friends in Edinburgh and elsewhere I also gratefully acknowledge the supp ort of a SERCEPSRC research studentship v Contents Introduction Thermo dynamics Statistical Mechanics Free Energy and Phase Transitions Phase Transitions The Ising Mo del Statistical Mechanics OLattice Systems Calculation in Statistical Mechanical Problems Analytic Metho ds MonteCarlo Simulation MonteCarlo Simulation at Phase Transitions Discussion Review IntegrationPerturbation Metho ds Thermo dynamic Integration Multistage Sampling The Acceptance Ratio Metho d Mons FiniteSize metho d Widoms ParticleInsertion Metho d Histogram Metho ds NonCanonical Metho ds Umbrella Sampling Multicanonical Ensemble The Expanded Ensemble vi Valleaus DensityScaling Monte Carlo The Dynamical Ensemble Grand Canonical MonteCarlo The Gibbs Ensemble Other Metho ds Coincidence Counting Lo cal States Metho ds Rickman and Philp ots Metho ds The Partitioning Metho d of Bhanot et al Discussion Multicanonical and Related Metho ds Introduction The Multicanonical Distribution over Energy Macrostates An AlternativeThe Ground State Metho d The Multicanonical Distribution over Magnetisation Macrostates Techniques for Obtaining and Using the Multicanonical Ensemble Metho ds Using Visited States Incorp orating Prior Information Metho ds Using Transitions FiniteSize Scaling Using Transitions for Final Estimators Parallelism and Equilibration Results Free Energy and Canonical Averages of the d Ising Mo del A Comparison Between the Multicanonical Ensemble and Thermo dy namic Integration P M at c Beyond Multicanonical Sampling The Multicanonical and Expanded Ensembles The Random Walk Problem Optimal Sampling Use of the Transition Matrix Prediction of the Optimal Distribution vii Discussion A Study of an Isostructural Phase Transition Introduction Comparison of Thermo dynamic Integration and the Expanded EnsembleUse of an Einstein Solid Reference System Thermo dynamic Integration Expanded Ensemble with Einstein Solid Reference System Other Issues Direct Metho dMulticanonical Ensemble with Variable V The Multicanonical N pT Ensemble and its Implementation The Pathological Nature of the SquareWell System Finding the Preweighting Function The Pro duction Stage Canonical Averages FiniteSize Scaling and the Interfacial Region Mapping the Co existence Curve The Physical Basis of the Phase Transition Discussion Conclusion A Exact FiniteSize Scaling Results for the Ising Mo del B The DoubleTangent Construction C Statistical Errors and Correlation Times D Jackknife Estimators E Details of the SquareWell Solid Simulation viii Chapter Introduction We b egin by giving necessary background to the work carried out in this thesis We shall deal with the thermo dynamical and statistical mechanical notions that underpin our understanding of phase transitions in particular the ideas of entropy and free energy We shall describ e the role of computer simulation esp ecially MonteCarlo simulation and explain why the measurement of free energy presents particular challenges Thermo dynamics Statistical Mechanics Free Energy and Phase Transitions Who could ever calculate