The stochastic Gross-Pitaevskii equation: II

We provide a derivation of a more accurate version of the stochastic Gross-Pitaevskii equation, as introduced by Gardiner et al (2002 J. Phys. B: At. Mol. Opt. Phys. 35 1555). This derivation does not rely on the concept of local energy and momentum conservation and is based on a quasiclassical Wigner function representation of a 'high temperature' master equation for a Bose gas, which includes only modes below an energy cut-off ER that are sufficiently highly occupied (the condensate band). The modes above this cutoff (the non-condensate band) are treated as being essentially thermalized. The interaction between these two bands, known as growth and scattering processes, provides noise and damping terms in the equation of motion for the condensate band, which we call the stochastic Gross-Pitaevskii equation. This approach is distinguished by the control of the approximations made in its derivation and by the feasibility of its numerical implementation.

Rising longevity, education, savings, and growth

This paper examines the impact of declines in adult mortality on growth in an overlapping generations model. With public education and imperfect annuity markets, a decline in mortality affects growth through three channels. First, it raises the saving rate and thereby increases the rate of physical capital accumulation. Second, it reduces accidental bequests, lowers investment, and thereby lowers the rate of physical capital accumulation. Third, it may lead the median voter to increase the tax rate for public education initially but lower the tax rate in a later stage. Starting from a high mortality rate as found in many Third World populations, the net effect of a decline in mortality is to raise the growth rate. However, starting from a low mortality rate such as is found in most industrial populations, the net effect of a further decline in mortality is to reduce the growth rate. The findings appear consistent with recent empirical evidence. (C) 2002 Elsevier Science B.V All rights reserved.

How useful is the genuine savings rate as a sustainability indicator for regions within countries? Australia and Queensland compared

This article shows how macroeconomic indicators of sustainable development can be applied to the Queensland economy. While recognising the complex and contentious theoretical and practical issues in deriving the Genuine Savings Rate (GSR) to serve as such an indicator, we use the World Bank's methodology, which includes only mineral depletion, deforestation and carbon dioxide emissions as environmental terms, to estimate GSRs for Queensland for the period 1989 to 1999, and compare these to World Bank estimates of Australia's GSR for the same period. We find that Queensland has a higher rate of natural resource depletion and a lower GSR than the whole of Australia. We also examine how well the World Bank GSR performs as a 'headline' measure of overall sustainability, review criticisms of the GSR, and compare its implicit policy implications with those of net state savings, and of the GSR plus a suite of other indicators.

Projected Gross-Pitaevskii equation for harmonically confined Bose gases at finite temperature

We extend the projected Gross-Pitaevskii equation formalism of Davis [Phys. Rev. Lett. 87, 160402 (2001)] to the experimentally relevant case of thermal Bose gases in harmonic potentials and outline a robust and accurate numerical scheme that can efficiently simulate this system. We apply this method to investigate the equilibrium properties of the harmonically trapped three-dimensional projected Gross-Pitaevskii equation at finite temperature and consider the dependence of condensate fraction, position, and momentum distributions and density fluctuations on temperature. We apply the scheme to simulate an evaporative cooling process in which the preferential removal of high-energy particles leads to the growth of a Bose-Einstein condensate. We show that a condensate fraction can be inferred during the dynamics even in this nonequilibrium situation.

Properties of the stochastic Gross-Pitaevskii equation: finite temperature Ehrenfest relations and the optimal plane wave representation

We present Ehrenfest relations for the high temperature stochastic Gross-Pitaevskii equation description of a trapped Bose gas, including the effect of growth noise and the energy cutoff. A condition for neglecting the cutoff terms in the Ehrenfest relations is found which is more stringent than the usual validity condition of the truncated Wigner or classical field method-that all modes are highly occupied. The condition requires a small overlap of the nonlinear interaction term with the lowest energy single particle state of the noncondensate band, and gives a means to constrain dynamical artefacts arising from the energy cutoff in numerical simulations. We apply the formalism to two simple test problems: (i) simulation of the Kohn mode oscillation for a trapped Bose gas at zero temperature, and (ii) computing the equilibrium properties of a finite temperature Bose gas within the classical field method. The examples indicate ways to control the effects of the cutoff, and that there is an optimal choice of plane wave basis for a given cutoff energy. This basis gives the best reproduction of the single particle spectrum, the condensate fraction and the position and momentum densities.