66 research outputs found
Similarity solutions of a Becker-Döring system with time-dependent monomer input
We formulate the Becker-Döring equations for cluster growth in the presence of a time-dependent source of monomer input.
In the case of size-independent aggregation and ragmentation
rate coefficients we find similarity solutions which are approached in the large time limit. The form of the solutions depends on the rate of monomer input and whether fragmentation is present in the model; four distinct types of solution are found
Exact solutions for cluster-growth kinetics with evolving size and shape profiles
In this paper we construct a model for the simultaneous
compaction by which clusters are restructured, and growth
of clusters by pairwise coagulation. The model has the form
of a multicomponent aggregation problem in which the
components are cluster mass and cluster diameter.
Following suitable approximations, exact explicit solutions
are derived which may be useful for the verification of
simulations of such systems. Numerical simulations are
presented to illustrate typical behaviour and to show the
accuracy of approximations made in deriving the model.
The solutions are then simplified using asymptotic
techniques to show the relevant timescales of the kinetic
processes and elucidate the shape of the cluster distribution
functions at large times
Nonlinear breathing modes at a defect
Recent molecular dynamics (MD) simulations of Cubero et al (1999) of
a DNA duplex containing the 'rogue' base difluorotoluene (F) in place of a
thymine (T) base show that breathing events can occur on the nanosecond
timescale, whereas breathing events in a normal DNA duplex take place on the microsecond timescale.
The main aim of this paper is to analyse a nonlinear Klein-Gordon lattice
model of the DNA duplex including both nonlinear interactions between
opposing bases and a defect in the interaction at one lattice site;
each of which can cause localisation of energy.
Solutions for a breather mode either side of the defect are derived using
multiple-scales asymptotics and are pieced together across the defect to
form a solution which includes the effects of the nonlinearity and the defect.
We consider defects in the
inter-chain interactions and in the along chain interactions.
In most cases we find in-phase breather modes and/or out-of-phase
breather modes, with one case displaying a shifted mode
An introduction to mathematical models of coagulation-fragmentation processes: a discrete deterministic mean-field approach
We summarise the properties and the fundamental mathematical results
associated with basic models which describe
coagulation and fragmentation processes in a deterministic manner
and in which cluster size is a discrete quantity (an integer
multiple of some basic unit size).
In particular, we discuss Smoluchowski's equation for aggregation,
the Becker-Döring model of simultaneous aggregation and fragmentation,
and more general models involving coagulation and fragmentation
Discrete breathers in a two-dimensional Fermi-Pasta-Ulam lattice
Using asymptotic methods, we investigate whether discrete
breathers are supported by a two-dimensional Fermi-Pasta-Ulam lattice. A scalar (one-component) two-dimensional
Fermi-Pasta-Ulam lattice is shown to model the charge stored
within an electrical transmission lattice. A third-order multiple-scale analysis in the semi-discrete limit fails, since at this order, the lattice equations reduce to the (2+1)-dimensional cubic nonlinear Schrödinger (NLS) equation which does not support stable soliton solutions for the breather envelope. We therefore extend
the analysis to higher order and find a generalised
-dimensional NLS equation which incorporates higher order dispersive and nonlinear terms as perturbations. We find an ellipticity criterion for the wave numbers of the carrier wave. Numerical simulations suggest that both stationary and moving breathers are supported by the system. Calculations of the energy show the expected threshold behaviour whereby the energy of breathers does {\em not} go to zero with the amplitude; we find
that the energy threshold is maximised by stationary breathers, and becomes arbitrarily small as the boundary of the domain of ellipticity is approached
Shape of transition layers in a differential--delay equation
We use asymptotic techniques to describe the bifurcation from steady-state to a periodic solution in the singularly perturbed delayed logistic equation ?x?(t) = ?x(t)+ ? f(x(t ? 1)) with ? ? 1. The solution has the form of plateaus of approximately unit width separated by narrow transition layers. The calculation of the period two solution is complicated by the presence of delay terms in the equation for the transition layers, which induces a phase shift that has to be calculated as part of the solution. High order asymptotic calculations enable both the shift and the shape of the layers to be determined analytically, and hence the period of the solution. We show numerically that the form of transition layers in the four-cycles is similar to that of the two-cycle, but that a three-cycle exhibits different behaviour
Asymptotic approximations to travelling waves in the diatomic Fermi-Pasta-Ulam lattice
We construct high-order approximate travelling waves solutions of the diatomic Fermi-Pasta-Ulam lattice using asymptotic techniques which are valid for arbitrary mass ratios. Separately small amplitude ansatzs are made for the motion of the lighter and heavier particles, which are coupled The Fredholm alternative is used to derive consistency conditions, whose solution generates small amplitude expansions for both sets of particles
Coarse-graining and renormalisation group methods for the elucidation of the kinetics of complex nucleation and growth processes
We review our work on generalisations of the Becker-Doring model of cluster-formation as applied to nucleation theory, polymer growth kinetics, and the formation of upramolecular structures in colloidal chemistry. One valuable tool in analysing mathematical models of these systems has been the coarse-graining approximation which enables macroscopic models for observable quantities to be derived from microscopic ones. This permits assumptions about the detailed molecular mechanisms to be tested, and their influence on the large-scale kinetics of surfactant self-assembly to be elucidated. We also summarise our more recent results on Becker-Doring systems, notably demonstrating that cross-inhibition and autocatalysis can
destabilise a uniform solution and lead to a competitive environment in which some species flourish at the expense of others, phenomena relevant in models of the origins of life
Chiral polymerisation and the RNA world
The purpose of this paper is to review two mathematical models: one
for the formation of homochiral polymers from an originally chirally
symmetric system; and the other, to show how, in an RNA-world scenario,
RNA can simultaneously act both as information storage
and a catalyst for its own production. We note the similarities and
differences in chemical mechanisms present in the systems.
We review these two systems, analysing steady-states,
interesting kinetics and the stability of symmetric solutions.
In both systems we show that there are ranges of parameter values where
some chains increase their own concentrations faster than others
Asymptotic analysis of combined breather-kink modes in a Fermi-Pasta-Ulam chain
We find approximations to travelling breather solutions of the
one-dimensional Fermi-Pasta-Ulam (FPU) lattice. Both bright
breather and dark breather solutions are found. We find that the
existence of localised (bright) solutions depends upon the
coefficients of cubic and quartic terms of the potential energy,
generalising an earlier inequality derived by James [CR Acad Sci
Paris 332, 581, (2001)]. We use the method of multiple scales to
reduce the equations of motion for the lattice to a nonlinear
Schr{\"o}dinger equation at leading order and hence construct an
asymptotic form for the breather. We show that in the absence of
a cubic potential energy term, the lattice supports combined
breathing-kink waveforms. The amplitude of breathing-kinks can be
arbitrarily small, as opposed to traditional monotone kinks, which
have a nonzero minimum amplitude in such systems. We also present
numerical simulations of the lattice, verifying the shape and
velocity of the travelling waveforms, and confirming the
long-lived nature of all such modes
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