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Turbofan oil systems are used to provide lubrication and cooling in the engine . There is an increasing interest in oil system architectures which utilize electric pumps and/or valves to give optimized control of flows to individual oil chambers, leading to improved thermal management of oil and lubrication efficiency. The challenges here lie in the trade-off between increasing controllability and minimizing the addition of new components, which adds unwanted production and maintenance costs. This paper formulates the oil system architecture design as a constrained, multiobjective optimization problem. An architecture is described using a graph with nodes representing components and edges representing interconnections between components. A fixed set of nodes called the architecture template is provided as an input and the edges are optimized for a multicriteria objective function. A heuristic method for determining similarities between the different oil chamber flow requirements is presented. This is used in the optimization to evaluate the controllability objective based on the structure of the valve architecture. The methodology provides benefits to system designers by selecting cheaper architectures with fewer valves when the need to control oil chambers separately is small. The effect of manipulating the cost/controllability criteria weightings is investigated to show the impact on the resulting architecture.

Marine energy in the United Kingdom is undergoing a period of growth in terms of development and implementation. The current installed tidal energy capacity is expected to rise to provide 20% of the United Kingdom’s electricity demand by 2050. This article used life cycle assessment to study four tidal energy devices, comparing their embodied energy and carbon dioxide emissions. The device designs studied included a multi-blade turbine, two three-blade horizontal axis turbine machines and an Archimedes’ screw device. These machines were chosen to represent a cross section of design for the device, foundation, installation and operation. Embodied energy was considered over the lifetime of each device. Energy use from fabrication, transport, installation, maintenance, decommissioning and recycling was all calculated and compared to the energy generated by each device. Finally, the embodied energy, CO2 intensity and energy payback periods were compared to those of conventional power generating systems and other renewable energy sources. Devices were studied based on a functional unit, defined as a 10 MW array installed for 100 years. Of the devices studied, the OpenHydro Open Centre turbine was found to have the best ratio of generated to embodied energy. All devices achieved CO2 and energy payback within 12 years and exhibited CO2 intensity between 18 and 35 gCO2/kW h. This compares favourably against current energy sources such as wind (8–12 gCO2/kW h), solar photovoltaic (~30 gCO2/kW h), nuclear (~70 gCO2/kW h) and coal (~1000 gCO2/kW h).

The discovery of superfluidity in liquid helium-4 was announced to the scientific world on 8 January 1938, when two short papers were published back to back in Nature. One was by Peter Kapitza (Nature 141 74), the director of the Institute for Physical Problems in Moscow, and the other was by two young Canadian physicists, Jack Allen and Don Misener (Nature 141 75), both working at the Royal Society Mond Laboratory at the University of Cambridge in the UK. Both studies reported that liquid helium flowed with almost no measurable viscosity below the transition temperature of 2.18 K.

We investigate the superfluid carrier density ρ s in the BCS–BEC crossover regime of a Fermi superfluid at finite temperatures. Including pairing fluctuations within a Gaussian approximation, we calculate the superfluid order parameter, chemical potential, and ρ s in a consistent manner in the whole BCS–BEC crossover region. In the weak-coupling BCS regime, the temperature dependence of ρ s is dominated by single-particle excitations accompanied by thermal dissociation of Cooper pairs. In the strong-coupling BEC regime, we show that single-particle excitations become less dominant and thermal excitations of collective Bogoliubov mode become important for ρ s . We clarify how the weak-coupling BCS result smoothly changes into the strong-coupling BEC result, as one increases the strength of a pairing interaction.

Recently the authors used the Kadanoff–Baym non-equilibrium Green's function formalism to derive kinetic equation for the non-condensate atoms, in conjunction with a consistent generalization of the Gross–Pitaevskii equation for the Bose condensate wavefunction. This work was limited to high temperatures, where the excited atoms could be described by a Hartree–Fock particle-like spectrum. Following the approach of Kane and Kadanoff in 1965, we present the generalization of our recent work which is valid at low temperatures, where the input single-particle spectrum is now described by the Bogoliubov–Popov approximation. We derive a kinetic equation for the quasiparticle distribution function with collision integrals describing scattering between quasiparticles and the condensate atoms. From the general expression for the collision integral for the scattering between quasiparticle excitations, we find the quasiparticle distribution function corresponding to local equilibrium. This expression includes a quasiparticle chemical potential that controls the non-diffusive equilibrium between condensate atoms and the quasiparticle excitations. We derive a generalized Gross–Pitaevskii equation for the condensate wavefunction that also includes the damping effects due to collisions between atoms in the condensate and the thermally excited quasiparticles. For a uniform Bose gas, our kinetic equation for the thermally excited quasiparticles reduces to that found by Eckern, as well as by Kirkpatrick and Dorfman.

A review is given of recent theoretical work on the superfluid dynamics of trapped Bose gases at finite temperatures, where there is a significant fraction of non-condensate atoms. One can now reach large enough densities and collision cross-sections needed to probe the collective modes in the collision-dominated hydrodynamic region where the gas exhibits characteristic superfluid behavior involving the relative motions of the condensate and non-condensate components. The precise analogue of the Landau-Khalatnikov two-fluid hydrodynamic equations was recently derived from trapped Bose gases, starting from a generalized Gross-Pitaevskii equation for the condensate macroscopic wavefunction and a kinetic equation for the non-condensate atoms.

We extend our recent work on the two-fluid hydrodynamics of a Bose-condensed gas by including collisions involving both condensate and non-condensate atoms. These collisions are essential for establishing a state of local thermodynamic equilibrium between the condensate and non-condensate. Our theory is more general than the usual Landau two-fluid theory, to which it reduces in the appropriate limit, in that it allows one to describe situations in which a state of complete local equilibrium between the two components has not been reached. The exchange of atoms between the condensate and non-condensate is associated with a new relaxational mode of the gas., 4 pages, revtex, 1 postscript figure, Fig.1 has been corrected

Starting from an approximate microscopic model of a trapped Bose-condensed gas at finite temperatures, we derive an equation of motion for the condensate wavefunction and a quantum kinetic equation for the distribution function for the excited atoms. The kinetic equation is a generalization of our earlier work in that collisions between the condensate and non-condensate (C12) are now included, in addition to collisions between the excited atoms as described by the Uehling–Uhlenbeck (C22) collision integral. The continuity equation for the local condensate density contains a source term Γ12which is related to the C12collision term. If we assume that the C22collision rate is sufficiently rapid to ensure that the non-condensate distribution function can be approximated by a local equilibrium Bose distribution, the kinetic equation can be used to derive hydrodynamic equations for the non-condensate. The Γ12source terms appearing in these equations play a key role in describing the equilibration of the local chemical potentials associated with the condensate and non-condensate components. We give a detailed study of these hydrodynamic equations and show how the Landau two-fluid equations emerge in the frequency domain ωτμ ≪ τμis a characteristic relaxation time associated with C12collisions. More generally, the lack of complete local equilibrium between the condensate and non-condensate is shown to give rise to a new relaxational mode which is associated with the exchange of atoms between the two components. This new mode provides an additional source of damping in the hydrodynamic regime. Our equations are consistent with the generalized Kohn theorem for the center of mass motion of the trapped gas even in the presence of collisions. Finally, we formulate a variational solution of the equations which provides a very convenient and physical way of estimating normal mode frequencies. In particular, we use relatively simple trial functions within this approach to work out some of the monopole, dipole and quadrupole oscillations for an isotropic trap.

We present a systematic account of several approximations for the Beliaev self-energies for a uniform dilute Bose gas at finite temperature. We discuss the first-order Popov approximation, which gives a phonon velocity which vanishes as n 0 (T) , where n0(T) is the Bose condensate density. We generalize this Popov approximation using a t-matrix calculated with self-consistent ladder diagrams. This analysis shows that this t-matrix becomes very temperature-dependent and vanishes at Tc, in agreement with Bijlsma and Stoof (1995). Finally, we make a detailed study of the Beliaev–Popov (B–P) approximation for the self-energies which are second order in the t-matrix. We work out the contribution from each individual diagram and give formal expressions for the self-energies and the excitation energy spectrum valid at arbitrary temperature. Careful attention is given to all infrared-divergent contributions. We rederive the well-known T=0 result of Beliaev (1958) for long-wavelength excitations. The analogous evaluation of the finite-temperature B–P self-energies is given in the low-frequency and long-wavelength limit. The corrections to the chemical potential, excitation energy and damping are all found to be proportional to the temperature near Tc. All infrared divergent terms are shown to cancel out in physical quantities in the long-wavelength limit at arbitrary temperatures.

Griffin, Wu and Stringari have derived the hydrodynamic equations of a trapped dilute Bose gas above the Bose-Einstein transition temperature. We give the extension which includes hydrodynamic damping, following the classic work of Uehling and Uhlenbeck based on the Chapman-Enskog procedure. Our final result is a closed equation for the velocity fluctuations δvwhich includes the hydrodynamic damping due to the shear viscosity θ and the thermal conductivity κ. Following Kavoulakis, Pethick and Smith, we introduce a spatial cutoff in our linearized equations when the density is so low that the hydrodynamic description breaks down. Explicit expressions are given for θ and κ, which are position-dependent through dependence on the local fugacity when one includes the effect of quantum degeneracy of the trapped gas. We also discuss a trapped Bose-condensed gas, generalizing the work of Zaremba, Griffin and Nikuni to include hydrodynamic damping due to the (non-condensate) normal fluid.

We present a detailed study of the temperature dependence of the condensate and noncondensate density profiles of a Bose-condensed gas in a parabolic trap. These quantitites are calculated self-consistently using the Hartree-Fock-Bogoliubov equations within the Popov approximation. Below the Bose-Einstein transition the excitation frequencies have a realtively weak temperature dependence even though the condensate is strongly depleted. As the condensate density goes to zero through the transition, the excitation frequencies are strongly affected and approach the frequencies of a noninteracting gas in the high temperature limit., Comment: 4 pages, Latex, 4 postscript figures. Submitted to Physical Review Letters

The standard theoretical basis for understanding superfluidity in Bose systems was formulated by Beliaev in 1957, based on splitting the quantum field operator into a macroscopically occupied condensate component and a non-condensate component. This leads to a description of the condensate in terms of a ‘single-particle state’, the so-called macroscopic wavefunction. Since the discovery of Bose-condensed gases, an alternative theoretical picture has been developed which is based on a ‘coherent band’ of classically occupied states. This is often called the classical or c-field approach. The goal of this chapter is to review the differences between the Beliaev broken symmetry and c-field approach, and to argue that the c-field concept of a coherent condensate band of states has problems as a description of Bose superfluidity. However, the c-field idea of treating the lowest energy excitations classically can be used to advantage to simplify calculations within the Beliaev broken-symmetry formalism.

We quantize the recent hydrodynamic analysis of Stringari for the low-energy collective modes of a trapped Bose gas at $T=0$. This is based on the time-dependent Gross-Pitaevskii equation, but omits the kinetic energy of the density fluctuations. We diagonalize the hydrodynamic Hamiltonian in terms of the normal modes associated with the amplitude and phase of the inhomogeneous Bose order parameter. These normal modes provide a convenient basis for calculating observable quantities. As applications, we calculate the depletion of the condensate at $T=0$ as well as the inelastic light-scattering cross section $S({\bf q},\omega)$ from low-energy condensate fluctuations. The latter involves a sum over all normal modes, with a weight proportional to the square of the ${\bf q}$ Fourier component of the density fluctuation associated with a given mode. Finally, we show how the Thomas-Fermi hydrodynamic description can be derived starting from the coupled Bogoliubov equations., Comment: 25 pages, 4 figures, submitted to Phys. Rev. A

We study the gap-function anisotropy and collective mode spectrum ({omega}{le}2{Delta}) of the Anderson-Chakravarty model for high-{ital T}{sub {ital c}} oxides based on Cooper-pair tunneling in a bilayer (with strength {ital T}{sub {ital J}}). For both {ital s}-wave and {ital d}-wave pairing in the layers, the shape of the gap around the Fermi surface is strongly dependent on {ital T}{sub {ital J}}. Besides the usual Anderson-Bogoliubov phase and Littlewood-Varma amplitude modes, we find branches (optical phononlike modes) involving fluctuations of the relative phase and amplitude of the order parameters of the two layers. These modes are the dynamic signature of the Cooper-pair tunneling model since their energy and damping depends critically on the relative magnitudes of the Cooper-pair tunneling strength ({ital T}{sub {ital J}}) and the pairing interaction ({ital g}), but not significantly on the symmetry of the pairing. The generalization to a trilayer system (which can arise in Bi, Tl, and Hg copper oxides) is briefly discussed.

In 1970, Ruvalds and Zawadowski (RZ) outlined a microscopic theory of the two-roton pair excitation spectrum in superfluid4He. They pointed out that because of a Base condensate-induced coupling into the density fluctuation spectrum, these pair excitations show up in inelastic neutron data for S(Q, ω) as high energy multiparticle scattering above the maxon-roton quasiparticle peak. Stimulated by recent high-resolution neutron data from ILL, we have carried out an extensive study of the single-particle ϱ1(Q, ω) and the two-particle ϱ2(Q, ω) spectral densities within the RZ scenario, over a wide range of wavevectors (1 ≲Q ≲ 3 A−1), frequencies and temperatures. We extend the original RZ analysis (which concentrated on the two-roton spectrum) to include the multiparticle structure associated with the maxonroton and maxon-maxon spectra and present numerical results for both attractive and repulsive quasiparticle interactions. We also point out that the microscopic theory of S(Q, ω) in a Bose-condensed fluid shows that it involves a weighted sum of both ϱ1(Q, ω)and ϱ2(Q, ω). As a result, multiparticle structure exhibited by S(Q, ω) is not easily related to theoretical results for ϱ1(Q, ω) and ϱ2(Q, ω). Previous attempts in the literature to fit neutron data for S(Q, ω) to either ϱ1(Q, ω) or ϱ2(Q, ω) would not appear to have much quantitative basis, especially in the two-roton frequency region ω∼2ΔR.

In Bose supefluids, the single-particle Green's function can be directly related to the superfluid velocity-velocity correlation function in the hydrodynamic regime. An explicit expression for the single-particle spectral density was originally written down by Hohenberg and Martin in 1965, starting from the two-fluid equations for a superfluid. We give a simple derivation of their results. Using these results, we calculate the relative weights of first and second sound modes in the single-particle spectral density as a function of temperature in a uniform Bose gas. We show that the second sound mode makes a dominant contribution to the single-particle spectrum in relatively high temperature region. We also discuss the possibility of experimental observation of the second sound mode in a Bose gas by photoemission spectroscopy., 16 pages, 8 figures

Surprisingly, it was 30 years after the first liquefaction of (4)He in 1908 that the discovery that liquid (4)He is not just a 'cold' liquid was made. Below T = 2.18 K, it is a 'quantum' liquid which exhibits spectacular macroscopic quantum behaviour that can be seen with the naked eye. Since the observation of superfluidity in liquid (4)He is one of the greatest discoveries in modern physics, we present a day-to-day chronology of the tangled events which preceded the seminal discovery of zero viscosity in 1938 by Kapitza in Moscow and by Allen and Misener in Cambridge. On the theory side, London argued in 1938 that the microscopic basis for this new superfluid phase was the forgotten phenomenon of Bose-Einstein condensation (BEC) first suggested by Einstein in 1925. In 1941, Landau developed a very successful theory of superfluid (4)He, but it was not anchored in a microscopic theory of interacting atoms. It took another 20 years for theorists to unify the two seemingly different theories of Landau and London. Experiments on trapped superfluid atomic gases since 1995 have shone new light on superfluid (4)He. In the mid-1930s, London had emphasized that superconductivity in metals and superfluidity in liquid (4)He were similar. Experiments on trapped two-component Fermi gases in the last five years have shown that a Bose condensate is indeed the basis of both of these superfluid phases. This confirms the now famous Bardeen-Cooper-Schrieffer-BEC crossover scenario developed for superfluidity by Leggett and Nozières in the early 1980s but largely ignored until a few years ago. The study of superfluid (4)He will increasingly overlap with strongly interacting dilute quantum gases, perhaps opening up a new era of research on this most amazing liquid.

The quasiparticle lifetime due to scattering from antiparamagnons is investigated in the superconducting state. In the normal state, antiparamagnon scattering gives rise to a linear resistivity with temperature. Below T c , the antiparamagnon spectrum is suppressed as the quasiparticles, and hence the particle-hole antiparamagnons, condense out. This suppression leads to an increased quasiparticle lifetime, thereby increasing the normal fluid component of the conductivity σ n below T c

The discovery of Bose Einstein condensation (BEC) in trapped ultracold atomic gases in 1995 has led to an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first treatment of BEC at finite temperatures, this book presents a thorough account of the theory of two-component dynamics and nonequilibrium behaviour in superfluid Bose gases. It uses a simplified microscopic model to give a clear, explicit account of collective modes in both the collisionless and collision-dominated regions. Major topics such as kinetic equations, local equilibrium and two-fluid hydrodynamics are introduced at an elementary level. Explicit predictions are worked out and linked to experiments. Providing a platform for future experimental and theoretical studies on the finite temperature dynamics of trapped Bose gases, this book is ideal for researchers and graduate students in ultracold atom physics, atomic, molecular and optical physics and condensed matter physics.

In this chapter we describe the numerical methods that can be used to solve the ZNG equations in the context of a dynamical simulation. These equations consist of a generalized GP equation (3.21) for the condensate and a Boltzmann equation (3.42) for the thermal component. The fact that the two equations are coupled makes their numerical solution more complex than when either is considered on its own. Indeed, the distinct quantum and classical aspects of the problem require specifically tailored numerical methods. Although most of these methods are well established and described elsewhere (Taha and Ablowitz, 1984; Sanz-Serna and Calvo, 1994), we provide in this chapter a detailed pedagogical discussion that will serve as a guide to those interested in carrying out such calculations for trapped Bose gases. This chapter is based on the papers of Jackson and Zaremba (2002a,b). There are two main parts to the numerical problem. The first is developing a method for solving the time-dependent GP equation for an arbitrary three-dimensional geometry. This we take up in Section 11.1. Second, a method is needed for solving the Boltzmann equation that accounts for the dynamics of the thermal component. Here one must deal both with the Hamiltonian dynamics of the thermal atoms, as they move in the self-consistent mean field of the condensate and thermal cloud, and with the collisions that take place between the thermal atoms themselves (the C 22 collisions) and between the thermal atoms and the condensate (the C 12 collisions). The methods used to account for these two distinct collisional processes are taken up in Section 11.3. As we shall see, collisions play an important role and cannot be neglected even when the dynamical behaviour is dominated by mean-field interactions.

In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamovic-Tomasovic (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3. Overview of Green's function approach To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry. The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.

Using a variational approach, we solve the equations of two-fluid hydrodynamics for a uniform and trapped Fermi gas at unitarity. In the uniform case, we find that the first and second sound modes are remarkably similar to those in superfluid Helium, a consequence of strong interactions. In the presence of harmonic trapping, first and second sound become degenerate at certain temperatures. At these points, second sound hybridizes with first sound and is strongly coupled with density fluctuations, giving a promising way of observing second sound. We also discuss the possibility of exciting second sound by generating local heat perturbations., Comment: Published version. 7 pages, 4 figures

We present a novel interpretation of the traditional Landau-Feynman picture of excitations in superfluid $^{4}\mathrm{He}$. Detailed consideration of neutron-scattering data on the temperature dependence of S(Q,\ensuremath{\omega}) in the region 0.4\ensuremath{\lesssim}Q\ensuremath{\lesssim}2 A${\mathrm{\r{}}}^{\mathrm{\ensuremath{-}}1}$ strongly suggests that while the photon is a collective zero-sound mode, the maxon roton is a strongly renormalized single-particle excitation. The condensate-induced hybridization of these two modes explains the smooth photon-maxon-roton dispersion curve as well as the vanishing intensity of the maxon-roton peak in S(Q,\ensuremath{\omega}) above ${\mathit{T}}_{\ensuremath{\lambda}}$.

In the strong-coupling BEC region where a Feshbach resonance gives rise to tightly-bound dimer molecules, we show that a spin-polarized Fermi superfluid reduces to a simple Bose-Fermi mixture of Bose-condensed dimers and the leftover unpaired fermions. Using a many-body functional integral formalism, the Gaussian fluctuations give rise to an induced dimer-dimer interaction mediated by the unpaired fermions, with the dimer-fermion vertex being given by the (mean-field) Born approximation. Treating the pairing fluctuations to quartic order, we show how the action for a spin-polarized Fermi superfluid reduces to one for a Bose-Fermi mixture. This Bose-Fermi action includes an expression for the effective dimer-unpaired fermion interaction in a spin-polarized Fermi superfluid beyond the Born approximation, in the superfluid phase at finite temperatures. In the low-density limit, we show how this dimer-fermion interaction gives the s-wave scattering length $a_{BF}=1.18a_F$ ($a_F$ is the s-wave fermion scattering length), a result first derived by Skorniakov and Ter-Martirosian in 1957 for three interacting fermions., The expurgated version; accepted for publication in PRA

We present the results of a variational calculation of the frequencies of the low-lying Landau two-fluid hydrodynamic modes in a trapped Fermi superfluid gas at unitarity. Landau's two-fluid hydrodynamics is expected to be the correct theory of Fermi superfluids at finite temperatures close to unitarity, where strong interactions give rise to collisional hydrodynamics. Two-fluid hydrodynamics predicts the existence of in-phase modes in which the superfluid and normal fluid components oscillate together, as well as out-of-phase modes where the two components move against each other. We prove that at unitarity, the dipole and breathing in-phase modes are locally isentropic. Their frequencies are independent of temperature and are the same above and below the superfluid transition. The out-of-phase modes, in contrast, are strongly dependent on temperature and hence, can be used to test the thermodynamic properties and superfluid density of a Fermi gas at unitarity. We give numerical results for the frequencies of these new modes as function of temperature in an isotropic trap at unitarity., Comment: 16 pages; 5 figures

The superfluid density is a fundamental quantity describing the response to a rotation as well as in two-fluid collisional hydrodynamics. We present extensive calculations of the superfluid density \rho_s in the BCS-BEC crossover regime of a uniform superfluid Fermi gas at finite temperatures. We include strong-coupling or fluctuation effects on these quantities within a Gaussian approximation. We also incorporate the same fluctuation effects into the BCS single-particle excitations described by the superfluid order parameter \Delta and Fermi chemical potential \mu, using the Nozi\`eres and Schmitt-Rink (NSR) approximation. This treatment is shown to be necessary for consistent treatment of \rho_s over the entire BCS-BEC crossover. We also calculate the condensate fraction N_c as a function of the temperature, a quantity which is quite different from the superfluid density \rho_s. We show that the mean-field expression for the condensate fraction N_c is a good approximation even in the strong-coupling BEC regime. Our numerical results show how \rho_s and N_c depend on temperature, from the weak-coupling BCS region to the BEC region of tightly-bound Cooper pair molecules. In a companion paper by the authors (cond-mat/0609187), we derive an equivalent expression for \rho_s from the thermodynamic potential, which exhibits the role of the pairing fluctuations in a more explicit manner., Comment: 32 pages, 12 figures