Free Energy of a Dilute Bose Gas: Lower Bound

A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $\rho$ and temperature $T$. In the dilute regime, i.e., when $a^3\rho \ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term $4\pi a (2\rho^2 - [\rho-\rho_c]_+^2)$. Here, $\rho_c(T)$ denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and $[ ]_+$ denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., $T \sim \rho^{2/3}$ or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [arXiv:math-ph/0601051] for estimating correlations to temperatures below the critical one.Comment: LaTeX2e, 53 page

Free Energies of Dilute Bose gases: upper bound

We derive a upper bound on the free energy of a Bose gas system at density $\rho$ and temperature $T$. In combination with the lower bound derived previously by Seiringer \cite{RS1}, our result proves that in the low density limit, i.e., when $a^3\rho\ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\Delta f$ the free energy difference per volume between interacting and ideal Bose gases is equal to 4\pi a (2\rho^2-[\rho-\rhoc]^2_+). Here, \rhoc(T) denotes the critical density for Bose-Einstein condensation (for the ideal gas), and $[\cdot ]_+$ $=$ $\max\{\cdot, 0\}$ denotes the positive part.Comment: 56 pages, no figure

Phylogenetic relationships of African Caecilians (Amphibia: Gymnophiona): insights from mitochondrial rRNA gene sequences

Africa (excluding the Seychelles) has a diverse caecilian fauna, including the endemic family Scolecomorphidae and six endemic genera of the more cosmopolitan Caeciliidae. Previous molecular phylogenetic studies have not included any caecilians from the African mainland. Partial 12S and 16S mitochondrial gene sequences were obtained for two species of the endemic African Scolecomorphidae and five species and four genera of African Caeciliids, aligned against previously reported sequences for 16 caecilian species, and analysed using parsimony, maximum likelihood, Bayesian and distance methods. Results are in agreement with traditional taxonomy in providing support for the monophyly of the African Caeciliid genera Boulengerula and Schistometopum and for the Scolecomorphidae. They disagree in indicating that the Caeciliidae is paraphyletic with respect to the Scolecomorphidae. Although more data from morphology and/or molecules will be required to resolve details of the interrelationships of the African caecilian genera, the data provide strong support for at least two origins of caecilians in which the eye is reduced and covered with bone, and do not support the hypotheses that the caecilian assemblages of Africa, and of East and of West Africa are monophyletic

Developing specialist leaders of education: a research engagement approach

There has been little research to date on the continuing professional development needs of the several thousand Specialist Leaders of Education (SLE) now designated by the National College for Teaching and Leadership in England to work across schools as consultants on school-to-school support. This case study reports on the second and third stages of a four-stage research process designed to address these needs. The fi rst stage reported on the creation of a professional devel- opment framework for SLE ’ s using consultancy research. These middle stages test out this framework with a stakeholder group of SLEs, head- teachers and broker in a Teaching Schools Alliance. The fourth stage will track the implementation of professional development activities arising from these fi ndings. Apart from the speci fi c needs of SLE, this study will have wider relevance for all practitioners and researchers working in and with schools on leadership development using Research Engagement strategies and Joint Practice Development approaches in a so-called ‘ self- improving ’ school system

Investigation into the mechanisms by which microwave heating enhances separation of water-in-oil emulsions

The separation of water-in-oil emulsions made with Azeri crude was investigated using natural gravity settling and microwave heating techniques. Separation times could be reduced by an order of magnitude compared with untreated emulsions. Increasing the salinity of the water phase leads to a 15% average decrease in the settling time for untreated emulsions compared with over 90% for microwave-heated emulsions. An image analysis technique showed that the observed increases in settling time could not be attributed to changes in viscosity alone. Significant coalescence of water droplets occurs during microwave heating, however the effects of coalescence and viscosity reduction cannot be completely decoupled. Despite this, it is clear that it is the thermal effect of microwave heating that leads to improvements in settling times, and that any advantages in microwave heating over conventional heating can be explained by selective heating of the aqueous phase rather than so-called non-thermal effects

Drying Kinetics of Salt Solution Droplets:Water Evaporation Rates and Crystallization

Drying and crystallization of solution droplets is a problem of broad relevance, determining the microstructures of particles formed in spray-drying, the phase of particles delivered by, for example, aerosol formulations for inhalation therapies, and the impact of aerosols on radiative forcing and climate. The ephemeral nature of free droplets, particularly when considering the drying kinetics of droplets with highly volatile constituents, has often precluded the accurate measurement of transient properties such as droplet size and composition, preventing the robust assessment of predictive models of droplet-drying rates, nucleation, and crystallization. Here, we report novel measurements of the drying kinetics of individual aqueous sodium chloride solution droplets using an electrodynamic balance to isolate and trap single aerosol droplets (radius ≈ 25 μm). The initial solution droplet size and composition are shown to be highly reproducible in terms of drying rate and crystallization time when examined over hundreds of identical evaporating droplets. We introduce a numerical model that determines the concentration gradient across the radial profile of the droplet as it dries, considering both the surface recession because of evaporation and the diffusion of components within the droplet. Drying-induced crystallization is shown to be fully determined for this system, with nucleation and instantaneous crystallization occurring once a critical supersaturation level of 2.04 ± 0.02 is achieved at the surface of the evaporating droplet. This phenomenological model provides a consistent account of the timescale and surface concentration of free-droplet crystallization on drying for the different drying conditions studied, a necessary step in progress toward achieving control over rates of crystallization and the competitive formation of amorphous particles

Finite Temperature Properties of Quantum Antiferromagnets in a Uniform Magnetic Field in One and Two Dimensions

Consider a $d$-dimensional antiferromagnet with a quantum disordered ground state and a gap to bosonic excitations with non-zero spin. In a finite external magnetic field, this antiferromagnet will undergo a phase transition to a ground state with non-zero magnetization, describable as the condensation of a dilute gas of bosons. The finite temperature properties of the Bose gas in the vicinity of this transition are argued to obey a hypothesis of ZERO SCALE-FACTOR UNIVERSALITY for $d < 2$, with logarithmic violations in $d=2$. Scaling properties of various experimental observables are computed in an expansion in $\epsilon=2-d$, and exactly in $d=1$.Comment: 27 pages, REVTEX 3.0, 8 Postscript figures appended, YCTP-xyz

Quantum spin systems at positive temperature

We develop a novel approach to phase transitions in quantum spin models based on a relation to their classical counterparts. Explicitly, we show that whenever chessboard estimates can be used to prove a phase transition in the classical model, the corresponding quantum model will have a similar phase transition, provided the inverse temperature $\beta$ and the magnitude of the quantum spins \CalS satisfy \beta\ll\sqrt\CalS. From the quantum system we require that it is reflection positive and that it has a meaningful classical limit; the core technical estimate may be described as an extension of the Berezin-Lieb inequalities down to the level of matrix elements. The general theory is applied to prove phase transitions in various quantum spin systems with \CalS\gg1. The most notable examples are the quantum orbital-compass model on $\Z^2$ and the quantum 120-degree model on $\Z^3$ which are shown to exhibit symmetry breaking at low-temperatures despite the infinite degeneracy of their (classical) ground state.Comment: 47 pages, version to appear in CMP (style files included