First-principles quantum dynamics in interacting Bose gases II: stochastic gauges

First principles simulations of the quantum dynamics of interacting Bose gases using the stochastic gauge representation are analysed. In a companion paper, we showed how the positive-P representation can be applied to these problems using stochastic differential equations. That method, however, is limited by increased sampling error as time evolves. Here, we show how the sampling error can be greatly reduced and the simulation time significantly extended using stochastic gauges. In particular, local stochastic gauges (a subset) are investigated. Improvements are confirmed in numerical calculations of single-, double- and multi-mode systems in the weak-mode coupling regime. Convergence issues are investigated, including the recognition of two modes by which stochastic equations produced by phase-space methods in general can diverge: movable singularities and a noise-weight relationship. The example calculated here displays wave-like behaviour in spatial correlation functions propagating in a uniform 1D gas after a sudden change in the coupling constant. This could in principle be tested experimentally using Feshbach resonance methods.

Gaussian operator bases for correlated fermions

We formulate a general multi-mode Gaussian operator basis for fermions, to enable a positive phase-space representation of correlated Fermi states. The Gaussian basis extends existing bosonic phase-space methods to Fermi systems and thus allows first-principles dynamical or equilibrium calculations in quantum many-body Fermi systems. We prove the completeness of the basis and derive differential forms for products with one- and two-body operators. Because the basis satisfies fermionic superselection rules, the resulting phase space involves only c-numbers, without requiring anticommuting Grassmann variables. Furthermore, because of the overcompleteness of the basis, the phase-space distribution can always be chosen positive. This has important consequences for the sign problem in fermion physics.

A refined solution to Earth's hidden niobium: implications for evolution of continental crust and mode of core formation

New high-precision niobium (Nb) and tantalum (Ta) concentration data are presented for early Archaean metabasalts, metabasaltic komatiites and their erosion products (mafic metapelites) from SW Greenland and the Acasta gneiss complex, Canada. Individual datasets consistently show sub-chondritic Nb/Ta ratios averaging 15.1+/-11.6. This finding is discussed with regard to two competing models for the solution of the Nb-deficit that characterises the accessible Earth. Firstly, we test whether Nb could have sequestered into the core due to its slightly siderophile (or chalcophile) character under very reducing conditions, as recently proposed from experimental evidence. We demonstrate that troilite inclusions of the Canyon Diablo iron meteorite have Nb and V concentrations in excess of typical chondrites but that the metal phase of the Grant, Toluca and Canyon Diablo iron meteorites do not have significant concentrations of these lithophile elements. We find that if the entire accessible Earth Nb-deficit were explained by Nb in the core, only ca. 17% of the mantle could be depleted and that by 3.7 Ga, continental crust would have already achieved ca. 50% of its present mass. Nb/Ta systematics of late Archaean metabasalts compiled from the literature would further require that by 2.5 Ga, 90% of the present mass of continental crust was already in existence. As an alternative to this explanation, we propose that the average Nb/Ta ratio (15.1+/-11.6) of Earth's oldest mafic rocks is a valid approximation for bulk silicate Earth. This would require that ca. 13% of the terrestrial Nb resided in the Ta-free core. Since the partitioning of Nb between silicate and metal melts depends largely on oxygen fugacity and pressure, this finding could mean that metal/silicate segregation did not occur at the base of a deep magma ocean or that the early mantle was slightly less reducing than generally assumed. A bulk silicate Earth Nb/Ta ratio of 15.1 allows for depletion of up to 40% of the total mantle. This could indicate that in addition to the upper mantle, a portion of the lower mantle is depleted also, or if only the upper mantle were depleted, an additional hidden high Nb/Ta reservoir must exist. Comparison of Nb/Ta systematics between early and late Archaean metabasalts supports the latter idea and indicates deeply subducted high Nb/Ta eclogite slabs could reside in the mantle transition zone or the lower mantle. Accumulation of such slabs appears to have commenced between 2.5 and 2.0 Ga. Regardless of these complexities of terrestrial Nb/Ta systematics, it is shown that the depleted mantle Nb/Th ratio is a very robust proxy for the amount of extracted continental crust, because the temporal evolution of this ratio is dominated by Th-loss to the continents and not Nb-retention in the mantle. We present a new parameterisation of the continental crust volume versus age curve that specifically explores the possibility of lithophile element loss to the core and storage of eclogite slabs in the transition zone. (C) 2003 Elsevier Science B.V. All rights reserved.