On wave action and phase in the non-canonical Hamiltonian formulation

The long time–evolution of disturbances to slowly–varying solutions of partial differential equations is subject to the adiabatic invariance of the wave action. Generally, this approximate conservation law is obtained under the assumption that the partial differential equations are derived from a variational principle or have a canonical Hamiltonian structure. Here, the wave action conservation is examined for equations that possess a non–canonical (Poisson) Hamiltonian structure. The linear evolution of disturbances in the form of slowly varying wavetrains is studied using a WKB expansion. The properties of the original Hamiltonian system strongly constrain the linear equations that are derived, and this is shown to lead to the adiabatic invariance of a wave action. The connection between this (approximate) invariance and the (exact) conservation laws of pseudo–energy and pseudomomentum that exist when the basic solution is exactly time and space independent is discussed. An evolution equation for the slowly varying phase of the wavetrain is also derived and related to Berry's phase.

A comparative review of dimension reduction methods in approximate Bayesian computation

Approximate Bayesian computation (ABC) methods make use of comparisons between simulated and observed summary statistics to overcome the problem of computationally intractable likelihood functions. As the practical implementation of ABC requires computations based on vectors of summary statistics, rather than full data sets, a central question is how to derive low-dimensional summary statistics from the observed data with minimal loss of information. In this article we provide a comprehensive review and comparison of the performance of the principal methods of dimension reduction proposed in the ABC literature. The methods are split into three nonmutually exclusive classes consisting of best subset selection methods, projection techniques and regularization. In addition, we introduce two new methods of dimension reduction. The first is a best subset selection method based on Akaike and Bayesian information criteria, and the second uses ridge regression as a regularization procedure. We illustrate the performance of these dimension reduction techniques through the analysis of three challenging models and data sets.

Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation

Many modern statistical applications involve inference for complex stochastic models, where it is easy to simulate from the models, but impossible to calculate likelihoods. Approximate Bayesian computation (ABC) is a method of inference for such models. It replaces calculation of the likelihood by a step which involves simulating artificial data for different parameter values, and comparing summary statistics of the simulated data with summary statistics of the observed data. Here we show how to construct appropriate summary statistics for ABC in a semi-automatic manner. We aim for summary statistics which will enable inference about certain parameters of interest to be as accurate as possible. Theoretical results show that optimal summary statistics are the posterior means of the parameters. Although these cannot be calculated analytically, we use an extra stage of simulation to estimate how the posterior means vary as a function of the data; and we then use these estimates of our summary statistics within ABC. Empirical results show that our approach is a robust method for choosing summary statistics that can result in substantially more accurate ABC analyses than the ad hoc choices of summary statistics that have been proposed in the literature. We also demonstrate advantages over two alternative methods of simulation-based inference.

A theoretical investigation of a-Fe2O3–Cr2O3 solid solutions

We have examined the thermodynamic stability of a-Fe2O3–Cr2O3 solid solutions as a function of temperature and composition, using a combination of statistical mechanics with atomistic simulation techniques based on classical interatomic potentials, and the addition of a model magnetic interaction Hamiltonian. Our calculations show that the segregation of the Fe and Cr cations is marginally favourable in energy compared to any other cation distribution, and in fact the energy of any cation configuration of the mixed system is always slightly higher than the combined energies of equivalent amounts of the pure oxides separately. However, the positive enthalpy of mixing is small enough to allow the stabilisation of highly disordered solid solutions at temperatures of B400 K or higher. We have investigated the degree of cation disorder and the effective cell parameters of the mixed oxide as functions of temperature and composition, and we discuss the effect of magnetic interactions and lattice vibrations on the stability of the solid solution.

Solutions of the fully compressible semi-geostrophic system

The fully compressible semi-geostrophic system is widely used in the modelling of large-scale atmospheric flows. In this paper, we prove rigorously the existence of weak Lagrangian solutions of this system, formulated in the original physical coordinates. In addition, we provide an alternative proof of the earlier result on the existence of weak solutions of this system expressed in the so-called geostrophic, or dual, coordinates. The proofs are based on the optimal transport formulation of the problem and on recent general results concerning transport problems posed in the Wasserstein space of probability measures.

Multiple stationary solutions of an irradiated slab

A mathematical model describing the heat budget of an irradiated medium is introduced. The one-dimensional form of the equations and boundary conditions are presented and analysed. Heat transport at one face of the slab occurs by absorption (and reflection) of an incoming beam of short-wave radiation with a fraction of this radiation penetrating into the body of the slab, a diffusive heat flux in the slab and a prescribed incoming heat flux term. The other face of the slab is immersed in its own melt and is considered to be a free surface. Here, temperature continuity is prescribed and evolution of the surface is determined by a Stefan condition. These boundary conditions are flexible enough to describe a range of situations such as a laser shining on an opaque medium, or the natural environment of polar sea ice or lake ice. A two-stream radiation model is used which replaces the simple Beer’s law of radiation attenuation frequently used for semi-infinite domains. The stationary solutions of the governing equations are sought and it is found that there exists two possible stationary solutions for a given set of boundary conditions and a range of parameter choices. It is found that the existence of two stationary solutions is a direct result of the model of radiation absorption, due to its effect on the albedo of the medium. A linear stability analysis and numerical calculations indicate that where two stationary solutions exist, the solution corresponding to a larger thickness is always stable and the solution corresponding to a smaller thickness is unstable. Numerical simulations reveal that when there are two solutions, if the slab is thinner than the smaller stationary thickness it will melt completely, whereas if the slab is thicker than the smaller stationary thickness it will evolve toward the larger stationary thickness. These results indicate that other mechanisms (e.g. wave-induced agglomeration of crystals) are necessary to grow a slab from zero initial thickness in the parameter regime that yields two stationary solutions.

Similarity solutions describing the melting of a mushy layer

A model of the melting of a mushy region in the absence of fluid flow is presented. Similarity solutions are obtained which are used to describe melting from a hot plate with and without the generation of a completely molten region. These solutions are extended to describe the melting of a mushy region in contact with a hot liquid. A significant feature of melting mushy regions is that the phase change occurs internally by dissolution. Our solutions for melting of a mushy region are used to investigate this internal phase change and are compared with the classical Neumann solutions for melting of a pure substance.

Analytical solutions of a minimal model of species migration in a bounded domain

A minimal model of species migration is presented which takes the form of a parabolic equation with boundary conditions and initial data. Solutions to the differential problem are obtained that can be used to describe the small- and large-time evolution of a species distribution within a bounded domain. These expressions are compared with the results of numerical simulations and are found to be satisfactory within appropriate temporal regimes. The solutions presented can be used to describe existing observations of nematode distributions, can be used as the basis for further work on nematode migration, and may also be interpreted more generally.

On the role of specific interactions in the diffusion of nanoparticles in aqueous polymer solutions

Understanding nanoparticle diffusion within non-Newtonian biological and synthetic fluids is essential in designing novel formulations (e.g., nanomedicines for drug delivery, shampoos, lotions, coatings, paints, etc.), but is presently poorly defined. This study reports the diffusion of thiolated and PEGylated silica nanoparticles, characterized by small-angle neutron scattering, in solutions of various water-soluble polymers such as poly(acrylic acid) (PAA), poly(Nvinylpyrrolidone) (PVP), poly(ethylene oxide) (PEO), and hydroxyethylcellulose (HEC) probed using NanoSight nanoparticle tracking analysis. Results show that the diffusivity of nanoparticles is affected by their dimensions, medium viscosity, and, in particular, the specific interactions between nanoparticles and the macromolecules in solution; strong attractive interactions such as hydrogen bonding hamper diffusion. The water-soluble polymers retarded the diffusion of thiolated particles in the order PEO > PVP > PAA > HEC whereas for PEGylated silica particles retardation followed the order PAA > PVP = HEC > PEO. In the absence of specific interactions with the medium, PEGylated nanoparticles exhibit enhanced mobility compared to their thiolated counterparts despite some increase in their dimensions.

Phase separation and surface segregation in ceria-zirconia solid solutions

Using a combination of density functional theory calculations and statistical mechanics, we show that a wide range of intermediate compositions of ceria – zirconia solid solutions are thermodynamically metastable with respect to phase separation into Ce-rich and Zr-rich oxides. We estimate that the maximum equilibrium concentration of Zr in CeO2 at 1373 K is ~2%, and therefore equilibrated samples with higher Zr content are expected to exhibit heterogeneity at the atomic scale. We also demonstrate that in the vicinity of the (111) surface, cation redistribution at high temperatures will occur with significant Ce enrichment of the surface, which we attribute to the more covalent character of Zr-O bonds compared to Ce-O bonds. Although the kinetic barriers for cation diffusion normally prevent the decomposition/segregation of ceria-zirconia solid solutions in typical catalytic applications, the separation behaviour described here can be expected to occur in modern three-way catalytic converters, where very high temperatures are reached.

Reversible state transition in nanoconfined aqueous solutions

Using molecular dynamics simulations, we find a reversible transition between the dispersion and aggregation states of solute molecules in aqueous solutions confined in nanoscale geometry, which is not observed in macroscopic systems. The nanoscale confinement also leads to a significant increase of the critical aggregation concentration (CAC). A theoretical model based on Gibbs free energy calculation is developed to describe the simulation results. It indicates that the reversible state transition is attributed to the low free energy barrier (of order kBT) in between two energy minima corresponding to the dispersion and aggregation states, and the enhancement of the CAC results from the fact that at lower concentrations the number of solute molecules is not large enough to allow the formation of a stable cluster in the confined systems.

Mathematical solutions for electricity networks in a low carbon future

Smart meters are becoming more ubiquitous as governments aim to reduce the risks to the energy supply as the world moves toward a low carbon economy. The data they provide could create a wealth of information to better understand customer behaviour. However at the household, and even the low voltage (LV) substation level, energy demand is extremely volatile, irregular and noisy compared to the demand at the high voltage (HV) substation level. Novel analytical methods will be required in order to optimise the use of household level data. In this paper we briefly outline some mathematical techniques which will play a key role in better understanding the customer's behaviour and create solutions for supporting the network at the LV substation level.