## 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]