40 research outputs found
Similarity solutions of a Becker–Döring system with time-dependent monomer input
http://fr-online.de/in_und_ausland/kultur_und_medien/feuilleton/?em_cnt=1688541&em_cnt_page=1 In der FR schäumt mit starken Worten Editionswissenschaftler Reuß, der uns bereits durch eine Suada gegen Open Access unliebsam auffiel ?s=reuß+roland gegen Googles Bibliothekspartnerschaft. Ein ziemlich dümmliches, teuilweise ahnungsloses Ressentiment, das eklatant verkennt, welche großartigen Möglichkeiten der Editionsphilologie sich durch die Volltextindizierung eröffnen. Dass im Prinzip alle Büch..
A hierarchical cluster system based on Horton-Strahler rules for river networks
We consider a cluster system in which each cluster is characterized
by two parameters: an \order" i; following Horton-Strahler's rules, and a
\mass" j following the usual additive rule. Denoting by ci;j (t) the concen-
tration of clusters of order i and mass j at time t; we derive a coagulation-
like ordinary di erential system for the time dynamics of these clusters.
Results about existence and the behaviour of solutions as t ! 1 are ob-
tained, in particular we prove that ci;j (t) ! 0 and Ni(c(t)) ! 0 as t ! 1;
where the functional Ni( ) measures the total amount of clusters of a given
xed order i: Exact and approximate equations for the time evolution of
these functionals are derived. We also present numerical results that sug-
gest the existence of self-similar solutions to these approximate equations
and discuss its possible relevance for an interpretation of Horton's law of
river number
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
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
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
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
Existence and stability analysis of finite 0--0 Josephson junctions
We investigate analytically and numerically a Josephson junction on finite
domain with two -discontinuity points characterized by a jump of in
the phase difference of the junction, i.e. a 0--0 Josephson junction. The
system is described by a modified sine-Gordon equation. We show that there is
an instability region in which semifluxons will be spontaneously generated.
Using a Hamiltonian energy characterization, it is shown how the existence of
static semifluxons depends on the length of the junction, the facet length, and
the applied bias current. The critical eigenvalue of the semifluxons is
discussed as well. Numerical simulations are presented, accompanying our
analytical results.Comment: Submitte
Necessary conditions for breathers on continuous media to approximate breathers on discrete lattices
We start by considering the sine-Gordon partial differential equation (PDE with an arbitrary perturbation. Using the method of Kuzmak-Luke, we investigate those conditions the perturbation must satisfy in order for a breather solution to be a valid leading-order asymptotic approximation to the perturbed problem. We analyse the cases of both stationary and moving breathers. As examples, we consider perturbing terms which include typical linear damping, periodic sinusoidal driving, and dispersion caused by higher order spatial derivatives. The motivation for this study is that the mathematical modelling of physical systems, often leads to the discrete sine-Gordon system of ODEs which are then approximated in the long wavelength limit by the continuous sine-Gordon PDE. Such limits typically produce fourth-order spatial derivatives as higher order correction terms. The new results show that the stationary breather solution is a consistent solution of both the quasi-continuum SG equation and the forced/damped SG system. However, the moving breather is only a consistent solution of the quasi- continuum SG equation and not the damped SG system
Monte–Carlo simulation of crystallization in single‐chain square‐well homopolymers
© 2020 Wiley Periodicals LLC We present Monte–Carlo (MC) simulations of the crystallization transition of single-chain square-well homopolymers, with a continuous description of monomer positions. For long chains with short-ranged interactions this system shows a strong configurational bottleneck, which makes it difficult to explore the whole configuration space. To surmount this problem we combine parallel tempering with a nonstandard choice of tempering levels, a bespoke biasing strategy and a method to map results between different temperatures. We verify that our simulations mix well when simulating chains of 128 and 256 beads. Our simulation approach resolves issues with reproducibility of MC simulations reported in prior work, particularly for the transition region between the expanded coil and crystalline region. We obtain highly reproducible results for both the free energy landscape and the inverse temperature, with low statistical noise. We outline a method to extract the free energy barrier, at any temperature, for any choice of order parameter, illustrating this technique by computing the free energy landscape as a function of the Steinhardt–Nelson order parameter for a range of temperatures