Motivated by experiments on bose atoms in traps which have attractive interactions (e.g. 7 Li), we consider two models which may be solved exactly. We construct the ground states subject to the constraint that the system is rotating with angular momentum proportional to the number of atoms. In a conventional system this would lead to quantised vortices; here, for attractive interactions, we find that the angular momentum is absorbed by the centre of mass motion. Moreover, the state is uncondensed and is an example of a 'fragmented' condensate discussed by Nozières and Saint James. The same models with repulsive interactions are fully condensed in the thermodynamic limit.One of the most novel aspects of the creation of Bose condensates with neutral atoms in traps is the possibility of observing a bose gas with attractive interactions (negative scattering lengths). The case of 7 Li has been studied both experimentally [1,2] and theoretically. Condensation has been predicted to be stable for a sufficiently small number of particles or sufficiently weak interactions [3,4]. The instability to collapse when these conditions are not obeyed has also been discussed by several authors [5][6][7][8][9].In this Letter we show, using two exactly soluble models, that there may be other possibilities for noncondensed states with attractive interactions. The states are the 'fragmented' condensates discussed by Nozières and Saint James [10] in the context of excitonic bose condensates. The possibility of such states emerges from the realisation [11] that it is the exchange interaction which causes bosons with repulsive interactions to condense into a single one-particle state, if there are several one-particle ground states. Conversely for attractive interactions, the exchange term is negative and may prefer 'fragmented' [10] condensation into more than one state if there is a degeneracy (or perhaps if the interactions are sufficiently strong). Kagan et al. [4] argue that trapped gases with sufficiently large negative scattering lengths are unstable to the formation of clusters using a somewhat different argument, but with the same physical origin.The two models we examine are: particles in a harmonic trap with L quanta of angular momenta with attractive interactions treated as a degenerate perturbation [12]; rotating particles in a harmonic trap interacting with harmonic interactions [13][14][15][16]. (Both of these cases have been of interest for fermions [12,14], where rotation is replaced by a magnetic field and the phenomena are related to the fractional quantum Hall effect.) Rotation is considered in both cases, partly because the non-rotating ground state, in the thermodynamic limit, is trivial in both cases (for different reasons) and partly because the response to rotation is characteristic of superfluidity in the system [17,18].Consider the two-dimensional Hamiltonianin the limit where the dimensionless coupling is weak, |η| ≪ 1, so that the contact interaction can be treated perturbatively. We will now determine the gr...

We calculate the order parameter and anisotropy (elongation) of the configurations of a nematic polymer in the nematic phase. At low temperatures we find exponentially rapid growth of chain dimensions as a function of inverse temperature. In the nematic direction the chain eventually adopts a rod-like state. The thermal activation of hairpins (abrupt reversals in chain directions) causes this behaviour. However, at even lower temperatures the deviation from rod-like alignment is governed by gentle meandering away from the mean field direction. We construct a Maier-Saupe mean field theory of the nematic-isotropic transition, calculating for long chains the transition temperature as a function of chain properties and predicting a universal jump in the order parameter at the transition, ASnL = f .

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SynopsisIn experiments lakes 223 (L223) and 302 South (L302S) in the Experimental Lakes Area in north-western Ontario, and Little Rock Lake (LRL) in northern Wisconsin, were progressively acidified with sulphuric acid from original pH values of 6.1–6.8 to 4.7–5.1. Although the lakes were at different locations with different physical settings and assemblages of plants and animals including fish, there were remarkable similarities in their responses, particularly in regard to biogeochemical processes and effects on biota at lower trophic levels.All three lakes generated an important part of their buffering capacity internally b\ the reduction of sulphate, and to a lesser extent by the reduction of nitrate. Alkalinity production increased as concentrations of biologically-active strong acid anions increased. Models relating the residence times of sulphate and nitrate to water renewal, or first-order kinetics, effectively predicted events.Acidification disrupted nitrogen cycling in all three lakes. Nitrification was inhibited in L223 and L302S, while in LRL, nitrogen fixation was greatly decreased at low pH.The phytoplankton communities of all three lakes were originally dominated by chrysophyceans and cryptophyceans. However acidification changed the dominant species and decreased diversity. Acidification tended to increase phytoplankton production and standing crop slightly, probably because light penetration was increased.Littoral zones of all three lakes became increasingly dominated by a few species of filamentous green algae, which created nuisance blooms by pH 5.6. Mats or clouds of algae changed the entire character of the littoral zone.Acidification of L223 and L302S caused the loss of several species of large benthic crustaceans as pH changed from 6 to 5.6. Large, acid-sensitive littoral crustaceans were absent from LRL before acidification, probably because the lake was already too acidic.As acidity increased, the dominance of cladocerans within zooplankton communities increased. Daphnia catawba appeared at pH values near 5.6 and became more abundant at lower pHs as the lakes were acidified. Its appearance coincided with a decline in other Daphnia species: another cladoceran, Bosmina longirostris, increased in the experimentally-acidified lakes as did Keratella taurocephala: they became the dominant rotifers. Several sensitive zooplankton species declined or disappeared as the lakes were acidified, most notably Daphnia galeata mendotae, Epischura lacustris, Diaptomus sicilis and Keratella cochlearis.The responses of different fish varied; they appeared to depend on the sensitivity of key organisms in the food chain. The ability of key fish species to reproduce was impaired as early as pH 5.8; their reproduction, except for yellow perch in LRL, had ceased at pH 5.0 in all the three lakes.Acidification consistently reduced the diversity and richness of species in taxonomic groups studied, these effects resulting from losses of species and the increased dominance of a few acidophilic taxa.Responses of experimentally-acidified lakes in north-western Ontario and atmospherically-acidified lakes in eastern Ontario were similar in most respects where records allowed comparisons to be made, notably in relation to biogeochemical processes and the disappearance of acid-sensitive biota.When the acidification of L223 was reversed, several biotic components recovered quickly. Fish resumed reproduction at pHs similar to those at which it ceased when the lake was being acidified. The condition of lake trout improved as a result of greatly increased populations of small fish, their prey. Many species of insects and crustaceans that had been extirpated by acidification returned. Assemblages of phytoplankton and chironomids have retained an acidophilic character, although their diversity during recovery is similar to that at comparable pHs during progressive acidification. As their chemistry recovered, atmospherically-acidified lakes in the Sudbury area were able to sustain recruitment by species offish, including lake trout and white sucker, with rapid increases in the diversity of invertebrate taxa. Results from both L223 and lakes near Sudbury suggest a rapid partial recovery of lacustrine communities when acidification is reversed.It is concluded that the experimental lakes responded similarly to acidification, and that experimental acidification can reliably indicate the effects of acidification attributable to acidic precipitation.

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We demonstrate that an extended Bose condensate can be stable in a random potential for a suitable weak-repulsive limit of a dense Bose gas, even though the non-interacting case is pathological. The condensate exists primarily because the interactions allow screening of the random potential. This may happen even when the chemical potential is in the Lifshitz tails of the single-particle case. Indeed, we argue that there are no Lifshitz tail states in our dense but weakly-interacting system. Using a number-phase representation, we calculate the increase in the depletion of the condensate with increasing randomness (at fixed density) which indicates the eventual destruction of the condensed phase-perhaps to a localized phase. The physical picture discussed should be relevant to the understanding of helium thin films.

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Surface water acidity is decreasing in large areas of Europe and North America in response to reductions in atmospheric S deposition, but the ecological responses to these water-quality improvements are uncertain. Biota are recovering in some lakes and rivers, as water quality improves, but they are not yet recovering in others. To make sense of these different responses, and to foster effective management of the acid rain problem, we need to understand 2 things: i) the sequence of ecological steps needed for biotic communities to recover; and ii) where and how to intervene in this process should recovery stall. Here our purpose is to develop conceptual frameworks to serve these 2 needs. In the first framework, the primarily ecological one, a decision tree highlights the sequence of processes necessary for ecological recovery, linking them with management tools and responses to bottlenecks in the process. These bottlenecks are inadequate water quality, an inadequate supply of colonists to permit establishment, and community-level impediments to recovery dynamics. A second, more management-oriented framework identifies where we can intervene to overcome these bottlenecks, and what research is needed to build the models to operationalize the framework. Our ability to assess the benefits of S emission reduction would be simplified if we had models to predict the rate and extent of ecological recovery from acidification. To build such models we must identify the ecological steps in the recovery process. The frameworks we present will advance us towards this goal.