121 research outputs found
Discrete breathers in a two-dimensional hexagonal Fermi-Pasta-Ulam lattice
We consider a two-dimensional Fermi-Pasta-Ulam (FPU) lattice with hexagonal
symmetry. Using asymptotic methods based on small amplitude ansatz, at third
order we obtain a reduction to a cubic nonlinear Schrodinger equation (NLS) for
the breather envelope. However, this does not support stable soliton solutions,
so we pursue a higher-order analysis yielding a generalised NLS, which includes
known stabilising terms. We present numerical results which suggest that
long-lived stationary and moving breathers are supported by the lattice. We
find breather solutions which move in an arbitrary direction, an ellipticity
criterion for the wavenumbers of the carrier wave, asymptotic estimates for the
breather energy, and a minimum threshold energy below which breathers cannot be
found. This energy threshold is maximised for stationary breathers, and becomes
vanishingly small near the boundary of the elliptic domain where breathers
attain a maximum speed. Several of the results obtained are similar to those
obtained for the square FPU lattice (Butt & Wattis, J Phys A, 39, 4955,
(2006)), though we find that the square and hexagonal lattices exhibit
different properties in regard to the generation of harmonics, and the isotropy
of the generalised NLS equation.Comment: 29 pages, 14 Figure
Renormalisation-theoretic analysis of non-equilibrium phase transitions I: The Becker-Doring equations with power law rate coefficients
We study in detail the application of renormalisation theory to models of
cluster aggregation and fragmentation of relevance to nucleation and growth
processes. We investigate the Becker-Dorging equations, originally formulated
to describe and analyse non-equilibrium phase transitions, and more recently
generalised to describe a wide range of physicochemical problems. In the
present paper we analyse how the systematic coarse-graining renormalisation of
the \BD system of equations affects the aggregation and fragmentation rate
coefficients. We consider the case of power-law size-dependent cluster rate
coefficients which we show lead to only three classes of system that require
analysis: coagulation-dominated systems, fragmentation-dominated systems and
those where coagulation and fragmentation are exactly balanced. We analyse the
late-time asymptotics associated with each class.Comment: 18 pages, to appear in J Phys A Math Ge
Long-Time Behaviour and Self-Similarity in a Coagulation Equation with Input of Monomers
For a coagulation equation with Becker-Doring type interactions and
time-independent monomer input we study the detailed long-time behaviour of
nonnegative solutions and prove the convergence to a self-similar function.Comment: 30 pages, 5 Figures, now published in Markov Processes and Related
Fields 12, 367-398, (2006
Modelling crystal aggregation and deposition\ud in the catheterised lower urinary tract
Urethral catheters often become encrusted with crystals of magnesium struvite and calcium phosphate. The encrustation can block the catheter, which can cause urine retention in the bladder and reflux into the kidneys. We develop a mathematical model to investigate crystal deposition on the catheter surface, modelling the bladder as a reservoir of fluid and the urethral catheter as a rigid channel. At a constant rate, fluid containing crystal particles of unit size enters the reservoir, and flows from the reservoir through the channel and out of the system. The crystal particles aggregate, which we model using Becker–Döring coagulation theory, and are advected through the channel, where they continue to aggregate and are deposited on the channel’s walls. Inhibitor particles also enter the reservoir, and can bind to the crystals, preventing further aggregation and deposition. The crystal concentrations are spatially homogeneous in the reservoir, whereas the channel concentrations vary spatially as a result of advection, diffusion and deposition. We investigate the effect of inhibitor particles on the amount of deposition. For all parameter values, we find that crystals deposit along the full length of the channel, with maximum deposition close to the channel’s entrance
Renormalisation-theoretic analysis of non-equilibrium phase transitions II: The effect of perturbations on rate coefficients in the Becker-Doring equations
We study in detail the application of renormalisation theory to models of
cluster aggregation and fragmentation of relevance to nucleation and growth
processes. In particular, we investigate the Becker-Doring (BD) equations,
originally formulated to describe and analyse non-equilibrium phase
transitions, but more recently generalised to describe a wide range of
physicochemical problems. We consider here rate coefficients which depend on
the cluster size in a power-law fashion, but now perturbed by small amplitude
random noise. Power-law rate coefficients arise naturally in the theory of
surface-controlled nucleation and growth processes. The noisy perturbations on
these rates reflect the effect of microscopic variations in such mean-field
coefficients, thermal fluctuations and/or experimental uncertainties. In the
present paper we generalise our earlier work that identified the nine classes
into which all dynamical behaviour must fall by investigating how random
perturbations of the rate coefficients influence the steady-state and kinetic
behaviour of the coarse-grained, renormalised system. We are hence able to
confirm the existence of a set of up to nine universality classes for such BD
systems.Comment: 30 pages, to appear in J Phys A Math Ge
Symmetry-breaking in chiral polymerisation
We propose a model for chiral polymerisation and investigate its symmetric
and asymmetric solutions. The model has a source species which decays into
left- and right-handed types of monomer, each of which can polymerise to form
homochiral chains; these chains are susceptible to `poisoning' by the opposite
handed monomer. Homochiral polymers are assumed to influence the proportion of
each type of monomer formed from the precursor. We show that for certain
parameter values a positive feedback mechanism makes the symmetric steady-state
solution unstable. The kinetics of polymer formation are then analysed in the
case where the system starts from zero concentrations of monomer and chains. We
show that following a long induction time, extremely large concentrations of
polymers are formed for a short time, during this time an asymmetry introduced
into the system by a random external perturbation may be massively amplified.
The system then approaches one of the steady-state solutions described above.Comment: 26pages, 6 Figure
Ink Drying in Inkjet Printers
The first problem put to the Study Group for Maths in Industry by Domino UK Ltd
concerns ink drying and blocking nozzles in a printer. The goals were as follows:
1. To propose mechanisms for the growth of a plug of dried ink in the open end of a Drop-on-Demand drop generator,
2. To suggest cures to this problem,
3. To consider why oscillating the meniscus appears to alleviate the problem
Stochastic Fractal and Noether's Theorem
We consider the binary fragmentation problem in which, at any breakup event, one of the daughter segments either survives with probability or disappears with probability . It describes a stochastic dyadic Cantor set that evolves in time, and eventually becomes a fractal. We investigate this phenomenon, through analytical methods and Monte Carlo simulation, for a generic class of models, where segment breakup points follow a symmetric beta distribution with shape parameter , which also determines the fragmentation rate. For a fractal dimension , we find that the -th moment is a conserved quantity, independent of and . We use the idea of data collapse -- a consequence of dynamical scaling symmetry -- to demonstrate that the system exhibits self-similarity. In an attempt to connect the symmetry with the conserved quantity, we reinterpret the fragmentation equation as the continuity equation of a Euclidean quantum-mechanical system. Surprisingly, the Noether charge corresponding to dynamical scaling is trivial, while relates to a purely mathematical symmetry: quantum-mechanical phase rotation in Euclidean time
Dissociation in a polymerization model of homochirality
A fully self-contained model of homochirality is presented that contains the
effects of both polymerization and dissociation. The dissociation fragments are
assumed to replenish the substrate from which new monomers can grow and undergo
new polymerization. The mean length of isotactic polymers is found to grow
slowly with the normalized total number of corresponding building blocks.
Alternatively, if one assumes that the dissociation fragments themselves can
polymerize further, then this corresponds to a strong source of short polymers,
and an unrealistically short average length of only 3. By contrast, without
dissociation, isotactic polymers becomes infinitely long.Comment: 16 pages, 6 figures, submitted to Orig. Life Evol. Biosp
Discrete breathers in honeycomb Fermi-Pasta-Ulam lattices
We consider the two-dimensional Fermi-Pasta-Ulam lattice with hexagonal honeycomb symmetry, which is a Hamiltonian system describing the evolution of a scalar-valued quantity subject to nearest neighbour interactions. Using multiple-scale analysis we reduce the governing lattice equations to a nonlinear Schrodinger (NLS) equation coupled to a second equation for an accompanying slow mode. Two cases in which the latter equation can be solved and so the system decoupled are considered in more detail: firstly, in the case of a symmetric potential, we derive the form of moving breathers. We find an ellipticity criterion for the wavenumbers of the carrier wave, together with asymptotic estimates for the breather energy. The minimum energy threshold depends on the wavenumber of the breather. We find that this threshold is locally maximised by stationary breathers. Secondly, for an asymmetric potential we find stationary breathers, which, even with a quadratic nonlinearity generate no second harmonic component in the breather. Plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure. Finally, we compare the properties of stationary breathers in the square, triangular and honeycomb lattices
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