The mathematical research of William Parry FRS

In this article we survey the mathematical research of the late William (Bill) Parry, FRS

Two Parameters for Three Dimensional Wetting Transitions

Critical effects at complete and critical wetting in three dimensions are studied using a coupled effective Hamiltonian H[s(y),\ell]. The model is constructed via a novel variational principle which ensures that the choice of collective coordinate s(y) near the wall is optimal. We highlight the importance of a new wetting parameter \Omega(T) which has a strong influence on critical properties and allows the status of long-standing Monte-Carlo simulation controversies to be re-examined.Comment: 4 pages RevTex, 2 encapsulated postscript figures, to appear in Europhys. Let

Towards a framework for analyzing interactions between social science and environmental policy

Interactions between social science and environmental policy have become increasingly important over the past 25 years. There has, however, been little analysis of the roles that social scientists adopt and the contributions they make. In this paper we begin the process, offering tentative answers to two key questions: in relation to environmental problems: (1) how do social science and public policy interact? and (2) in the future, what types of interactions can social scientists engage in? To answer these questions we build on research in policy studies and science and technology studies, and extend it through public scholarship debates

Interfacial Structural Changes and Singularities in Non-Planar Geometries

We consider phase coexistence and criticality in a thin-film Ising magnet with opposing surface fields and non-planar (corrugated) walls. We show that the loss of translational invariance has a strong and unexpected non-linear influence on the interface structure and phase diagram. We identify 4 non-thermodynamic singularities where there is a qualitative change in the interface shape. In addition, we establish that at the finite-size critical point, the singularity in the interface shape is characterized by two distint critical exponents in contrast to the planar case (which is characterised by one). Similar effects should be observed for prewetting at a corrugated substrate. Analogy is made with the behaviour of a non-linear forced oscillator showing chaotic dynamics.Comment: 13 pages, 3 figure

Corrugation-Induced First-Order Wetting: An Effective Hamiltonian Study

We consider an effective Hamiltonian description of critical wetting transitions in systems with short-range forces at a corrugated (periodic) wall. We are able to recover the results obtained previously from a `microscopic' density-functional approach in which the system wets in a discontinuous manner when the amplitude of the corrugations reaches a critical size A*. Using the functional renormalization group, we find that A* becomes dependent on the wetting parameter \omega in such a way as to decrease the extent of the first-order regime. Nevertheless, we still expect wetting in the Ising model to proceed in a discontinuous manner for small deviations of the wall from the plane.Comment: 9 pages RevTex with 2 EPS figures. To appear in Eur. Phys. J.

An exact solution for two dimensional wetting with a corrugated wall

An exact solution of a two dimensional RSOS model of wetting at a corrugated (periodic) wall is found using transfer matrix techniques. In contrast to mean-field analysis of the same problem the wetting transition remains second-order and occurs at a lower temperature than that of the planar system. Comparison with numerical studies and other analytical approaches is made.Comment: 11 pages LaTex with 1 eps figure. To appear in J.Phys.

A Comparison of Three Curve Intersection Algorithms

An empirical comparison is made between three algorithms for computing the points of intersection of two planar Bezier curves. The algorithms compared are: the well known Bezier subdivision algorithm, which is discussed in Lane 80; a subdivision algorithm based on interval analysis due to Koparkar and Mudur; and an algorithm due to Sederberg, Anderson and Goldman which reduces the problem to one of finding the roots of a univariate polynomial. The details of these three algorithms are presented in their respective references

Coupled Hamiltonians and Three Dimensional Short-Range Wetting Transitions

We address three problems faced by effective interfacial Hamiltonian models of wetting based on a single collective coordinate \ell representing the position of the unbinding fluid interface. Problems (P1) and (P2) refer to the predictions of non-universality at the upper critical dimension d=3 at critical and complete wetting respectively which are not borne out by Ising model simulation studies. (P3) relates to mean-field correlation function structure in the underlying continuum Landau model. We investigate the hypothesis that these concerns arise due to the coupling of order parameter fluctuations near the unbinding interface and wall. For quite general choices of collective coordinates X_i we show that arbitrary two-field models H[X_1,X_2] can recover the required anomalous structure of mean-field correlation functions (P3). To go beyond mean-field theory we introduce a set of Hamiltonians based on proper collective coordinates s near the wall which have both interfacial and spin-like components. We argue that an optimum model H[s,\ell] in which the degree of coupling is controlled by an angle-like variable, best describes the non-universality of the Ising model and investigate its critical behaviour. For critical wetting the appropriate Ginzburg criterion shows that the true asymptotic critical regime for the local susceptibility \chi_1 is dramatically reduced consistent with observations of mean-field behaviour in simulations (P1). For complete wetting the model yields a precise expression for the temperature dependence of the renormalized critical amplitude \theta in good agreement with simulations (P2). We highlight the importance of a new wetting parameter which describes the physics that emerges due to the coupling effects.Comment: 34 pages, RevTex, 8 eps figures. To appear in Physica