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The mathematical research of William Parry FRS
In this article we survey the mathematical research of the late William (Bill) Parry, FRS
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
Brane Gas Inflation
We consider the brane gas picture of the early universe. At later stages,
when there are no winding modes and the background is free to expand, we show
that a moving 3-brane, which we identify with our universe, can inflate even
though it is radiation-dominated. The crucial ingredients for successful
inflation are the coupling to the dilaton and the equation of state of the
bulk. If we suppose the brane initially forms in a collision of
higher-dimensional branes, then the spectrum of primordial density fluctuations
naturally has a thermal origin.Comment: 4 pages, 1 figur
Fluid adsorption near an apex: Covariance between complete and critical wetting
Critical wetting is an elusive phenomenon for solid-fluid interfaces. Using
interfacial models we show that the diverging length scales, which characterize
complete wetting at an apex, precisely mimic critical wetting with the apex
angle behaving as the contact angle. Transfer matrix, renormalization group
(RG) and mean field analysis (MF) shows this covariance is obeyed in 2D, 3D and
for long and short ranged forces. This connection should be experimentally
accesible and provides a means of checking theoretical predictions for critical
wetting.Comment: 4 pages, 1 figure, submitted to Physical Review Letter
Coupled Fluctuations near Critical Wetting
Recent work on the complete wetting transition has emphasized the role played
by the coupling of fluctuations of the order parameter at the wall and at the
depinning fluid interface. Extending this approach to the wetting transition
itself we predict a novel crossover effect associated with the decoupling of
fluctuations as the temperature is lowered towards the transition temperature
T_W. Using this we are able to reanalyse recent Monte-Carlo simulation studies
and extract a value \omega(T_W)=0.8 at T_W=0.9T_C in very good agreement with
long standing theoretical predictions.Comment: 4 pages, LaTex, 1 postscript figur
New bulk scalar field solutions in brane worlds
We use nonlinear perturbation theory to obtain new solutions for brane world
models that incorporate a massive bulk scalar field. We then consider tensor
perturbations and show that Newtonian gravity is recovered on the brane for
both a light scalar field and for a bulk field with large negative mass. This
latter result points to the viability of higher-derivative theories of gravity
in the context of bulk extra dimensions.Comment: 4+\epsilon pages, no figure
Correlation function algebra for inhomogeneous fluids
We consider variational (density functional) models of fluids confined in
parallel-plate geometries (with walls situated in the planes z=0 and z=L
respectively) and focus on the structure of the pair correlation function
G(r_1,r_2). We show that for local variational models there exist two
non-trivial identities relating both the transverse Fourier transform G(z_\mu,
z_\nu;q) and the zeroth moment G_0(z_\mu,z_\nu) at different positions z_1, z_2
and z_3. These relations form an algebra which severely restricts the possible
form of the function G_0(z_\mu,z_\nu). For the common situations in which the
equilibrium one-body (magnetization/number density) profile m_0(z) exhibits an
odd or even reflection symmetry in the z=L/2 plane the algebra simplifies
considerably and is used to relate the correlation function to the finite-size
excess free-energy \gamma(L). We rederive non-trivial scaling expressions for
the finite-size contribution to the free-energy at bulk criticality and for
systems where large scale interfacial fluctuations are present. Extensions to
non-planar geometries are also considered.Comment: 15 pages, RevTex, 4 eps figures. To appear in J.Phys.Condens.Matte
3D wedge filling and 2D random-bond wetting
Fluids adsorbed in 3D wedges are shown to exhibit two types of continuous
interfacial unbinding corresponding to critical and tricritical filling
respectively. Analytic solution of an effective interfacial model based on the
transfer-matrix formalism allows us to obtain the asymptotic probability
distribution functions for the interfacial height when criticality and
tricriticality are approached. Generalised random walk arguments show that, for
systems with short-ranged forces, the critical singularities at these
transitions are related to 2D complete and critical wetting with random bond
disorder respectively.Comment: 7 pages, 3 figures, accepted for publication in Europhysics Letter
Derivation of a Non-Local Interfacial Hamiltonian for Short-Ranged Wetting II: General Diagrammatic Structure
In our first paper, we showed how a non-local effective Hamiltionian for
short-ranged wetting may be derived from an underlying Landau-Ginzburg-Wilson
model. Here, we combine the Green's function method with standard perturbation
theory to determine the general diagrammatic form of the binding potential
functional beyond the double-parabola approximation for the
Landau-Ginzburg-Wilson bulk potential. The main influence of cubic and quartic
interactions is simply to alter the coefficients of the double parabola-like
zig-zag diagrams and also to introduce curvature and tube-interaction
corrections (also represented diagrammatically), which are of minor importance.
Non-locality generates effective long-ranged many-body interfacial interactions
due to the reflection of tube-like fluctuations from the wall. Alternative wall
boundary conditions (with a surface field and enhancement) and the diagrammatic
description of tricritical wetting are also discussed.Comment: (14 pages, 2 figures) Submitted J. Phys. Condens. Matte
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