This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

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Many-Body Physics with Ultracold Gases

We present an overview of the lattice Boltzmann method (LBM), a parallel and efficient algorithm for simulating single-phase and multiphase fluid flows and for incorporating additional physical complexities. The LBM is especially useful for modeling complicated boundary conditions and multiphase interfaces. Recent extensions of this method are described, including simulations of fluid turbulence, suspension flows, and reaction diffusion systems.

Lattice boltzmann method for fluid flows

The phenomenon of Bose-Einstein condensation of dilute gases in traps is reviewed from a theoretical perspective. Mean-field theory provides a framework to understand the main features of the condensation and the role of interactions between particles. Various properties of these systems are discussed, including the density profiles and the energy of the ground-state configurations, the collective oscillations and the dynamics of the expansion, the condensate fraction and the thermodynamic functions. The thermodynamic limit exhibits a scaling behavior in the relevant length and energy scales. Despite the dilute nature of the gases, interactions profoundly modify the static as well as the dynamic properties of the system; the predictions of mean-field theory are in excellent agreement with available experimental results. Effects of superfluidity including the existence of quantized vortices and the reduction of the moment of inertia are discussed, as well as the consequences of coherence such as the Josephson effect and interference phenomena. The review also assesses the accuracy and limitations of the mean-field approach.

https://arxiv.org/pdf/cond-mat/9806038.pdf

Theory of Bose-Einstein condensation in trapped gases

We propose the lattice BGK models, as an alternative to lattice gases or the lattice Boltzmann equation, to obtain an efficient numerical scheme for the simulation of fluid dynamics. With a properly chosen equilibrium distribution, the Navier-Stokes equation is obtained from the kinetic BGK equation at the second-order of approximation. Compared to lattice gases, the present model is noise-free, has Galileian invariance and a velocity-independent pressure. It involves a relaxation parameter that influences the stability of the new scheme. Numerical simulations are shown to confirm the speed of sound and the shear viscosity.

https://iopscience.iop.org/article/10.1209/0295-5075/17/6/001/pdf

Lattice BGK Models for Navier-Stokes Equation

The generalized hydrodynamics (the wave vector dependence of the transport coefficients) of a generalized lattice Boltzmann equation (LBE) is studied in detail. The generalized lattice Boltzmann equation is constructed in moment space rather than in discrete velocity space. The generalized hydrodynamics of the model is obtained by solving the dispersion equation of the linearized LBE either analytically by using perturbation technique or numerically. The proposed LBE model has a maximum number of adjustable parameters for the given set of discrete velocities. Generalized hydrodynamics characterizes dispersion, dissipation (hyper-viscosities), anisotropy, and lack of Galilean invariance of the model, and can be applied to select the values of the adjustable parameters which optimize the properties of the model. The proposed generalized hydrodynamic analysis also provides some insights into stability and proper initial conditions for LBE simulations. The stability properties of some 2D LBE models are analyzed and compared with each other in the parameter space of the mean streaming velocity and the viscous relaxation time. The procedure described in this work can be applied to analyze other LBE models. As examples, LBE models with various interpolation schemes are analyzed. Numerical results on shear flow with an initially discontinuous velocity profile (shock) with or without a constant streaming velocity are shown to demonstrate the dispersion effects in the LBE model; the results compare favorably with our theoretical analysis. We also show that whereas linear analysis of the LBE evolution operator is equivalent to Chapman-Enskog analysis in the long wave-length limit (wave vector k = 0), it can also provide results for large values of k. Such results are important for the stability and other hydrodynamic properties of the LBE method and cannot be obtained through Chapman-Enskog analysis.

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Theory of the lattice boltzmann method: dispersion, dissipation, isotropy, galilean invariance, and stability

A kinetic theory approach to collision processes in ionized and neutral gases is presented. This approach is adequate for the unified treatment of the dynamic properties of gases over a continuous range of pressures from the Knudsen limit to the high-pressure limit where the aerodynamic equations are valid. It is also possible to satisfy the correct microscopic boundary conditions. The method consists in altering the collision terms in the Boltzmann equation. The modified collision terms are constructed so that each collision conserves particle number, momentum, and energy; other characteristics such as persistence of velocities and angular dependence may be included. The present article illustrates the technique for a simple model involving the assumption of a collision time independent of velocity; this model is applied to the study of small amplitude oscillations of one-component ionized and neutral gases. The initial value problem for unbounded space is solved by performing a Fourier transformation on the space variables and a Laplace transformation on the time variable. For uncharged gases there results the correct adiabatic limiting law for sound-wave propagation at high pressures and, in addition, one obtains a theory of absorption and dispersion of sound for arbitrary pressures. For ionized gases the difference in the nature of the organization in the low-pressure plasma oscillations and in high-pressure sound-type oscillations is studied. Two important cases are distinguished. If the wavelengths of the oscillations are long compared to either the Debye length or the mean free path, a small change in frequency is obtained as the collision frequency varies from zero to infinity. The accompanying absorption is small; it reaches its maximum value when the collision frequency equals the plasma frequency. The second case refers to waves shorter than both the Debye length and the mean free path; these waves are characterized by a very heavy absorption.

A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems

Small amplitude processes in charged and neutral one-component systems

For a system of weakly repelling bosons, a theory of the elementary line vortex excitations is developed. The vortex state is characterised by the presence of a finite fraction of the particles in a single particle state of integer angular momentum. The radial dependence of the highly occupied state follows from a self-consistent field equation. The radial function and the associated particle density are essentially constant everywhere except inside a core, where they drop to zero. The core size is the de Broglie wavelength associated with the mean interaction energy per particle. The expectation value of the velocity has the radial dependence of a classical vortex. In this Hartree approximation the vorticity is zero everywhere except on the vortex line. When the description of the state is refined to include the zero point oscillations of the phonon field, the vorticity is spread out over the core. These results confirm in all essentials the intuitive arguments ofOnsager andFeynman. The phonons moving perpendicular to the vortex line are coherent excitations of equal and opposite angular momentum relative to the substratum of moving particles that constitute the vortex. The vortex motion resolves the degeneracy of the Bogoljubov phonons with respect to the azimuthal quantum number.

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Structure of a quantized vortex in boson systems

Van Vleck and Weisskopf and Fr\"ohlich have derived a microwave line shape by studying the interruption by collisions of the motion of a classical oscillator. They assume that after the instantaneous impact the oscillator variables are distributed according to a Boltzmann distribution appropriate to the value of the applied field at collision. In contrast to the earlier theory of Lorentz, they obtain the correct static polarization. The procedure involves an assumption of very large velocity during collision. This is criticized on the grounds that the duration of collision is short compared to the resonant period and energy exchanges are of the order of $\mathrm{kT}$. We have derived a line-shape formula assuming that the positions are unchanged after impact. Two extreme models are studied. In one, the oscillators have a Maxwellian distribution of velocities after impact; the second is a Brownian motion treatment. The resulting line shape in both cases is that of a friction-damped oscillator. For collision frequency much less than the resonant frequency, the polarization postulated by the above authors is reached as a result of kinematic motion between collisions, and the line shapes agree. However, to obtain equal line widths and peak absorptions, the collision frequency is twice as large for the present theory. For collision frequency comparable to resonant frequency a less distorted line shape results. For testing the theories, experiments on foreign-gas broadening in the microwave region at pressures of the order of an atmosphere are required. Differences between the theories are small for conditions accessible experimentally at present.

Shape of Collision-Broadened Spectral Lines

A theory of the small-amplitude oscillations of an ionized gas in a static magnetic field is developed, including the effects of temperature motions. The Boltzmann equation is solved for this problem, and exact expressions are obtained for the distribution function and dispersion relation. A general feature of the dispersion relation is the existence of gaps in the spectrum at frequencies which are approximately multiples of ${\ensuremath{\omega}}_{c}=\frac{\mathrm{eH}}{\mathrm{mc}}$. The magnitude of the gap depends on the temperature of the gas, being proportional to it for long wavelengths. This leads to the prediction of selective reflection of waves impinging on a plasma with frequency in the forbidden range.For $\mathrm{ck}\ensuremath{\gg}{\ensuremath{\omega}}_{p}$, ${\ensuremath{\omega}}_{c}$ the waves split into approximately longitudinal plasma waves and transverse waves. Detailed analysis is made of the plasma waves for ${\ensuremath{\omega}}_{c}$ small and ${\ensuremath{\omega}}_{c}$ large. At long wavelengths the frequency is ${\ensuremath{\omega}}^{2}\ensuremath{\simeq}\ensuremath{\omega}_{p}^{}{}_{}{}^{2}+\ensuremath{\omega}_{c}^{}{}_{}{}^{2}+\ensuremath{\beta}(\frac{\ensuremath{\kappa}T}{m}){k}^{2}$, where $\ensuremath{\beta}$ depends on ${\ensuremath{\omega}}_{p}$ and ${\ensuremath{\omega}}_{c}$. For waves near the Debye length the waves are heavily damped.Two simplified treatments of plasma oscillations based on transport equations are compared with the above treatment. Expressions of the form ${\ensuremath{\omega}}^{2}\ensuremath{\simeq}\ensuremath{\omega}_{p}^{}{}_{}{}^{2}+\ensuremath{\omega}_{c}^{}{}_{}{}^{2}+\ensuremath{\beta}(\frac{\ensuremath{\kappa}T}{m}){k}^{2}$ are obtained where the factor $\ensuremath{\beta}$ is independent of ${\ensuremath{\omega}}_{c}$ and ${\ensuremath{\omega}}_{p}$. In addition, the transport treatments fail to predict the heavy damping near the Debye length and the existence of gaps in the frequency spectrum.

E. P. Gross

Papers

A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems

Small amplitude processes in charged and neutral one-component systems

Structure of a quantized vortex in boson systems

Shape of Collision-Broadened Spectral Lines

Plasma Oscillations in a Static Magnetic Field