Dynamics of an Ultracold Bose Gas in Funnel-Shaped Potential

In this paper we develop a variational theory to study the dynamicproperties of ultracold Bose gas in a funnel external potential. We obtain one-dimensional nonlinear equation which describes the dynamics of transverse tight confined bosonic gas from three-dimension to one-dimension, and find one-dimensional s-wave scattering length which depends on the shape oftransverse confining potential. If the funnel trapping potential is strong enough at zero temperature, all transverse excitations are frozen. We find the dynamic equation which describes the Tonks-Girardeau gas and present a qualitative analysis of the experimental accessibility of the Tonks-Girardeau gas with funnel-trapped alkalic atoms.     

Dynamics of an Ultracold Bose Gas in Funnel-Shaped Potential

In this paper we develop a variational theory to study the dynamic properties of ultracold Bose gas in a funnel external potential. We obtain one-dimensional nonlinear equation which describes the dynamics of transverse tight confined bosonic gas from three-dimension to one-dimension, and find one-dimensional s-wave scattering length which depends on the shape of transverse confining potential. If the funnel trapping potential is strong enough at zero temperature, all transverse excitations are frozen. We find the dynamic equation which describes the Tonks-Girardeau gas and present a qualitative analysis of the experimental accessibility of the Tonks Girardeau gas with funnel-trapped alkalic atoms.     

Effective Dynamics of a Tracer Particle Interacting with an Ideal Bose Gas

We study a system consisting of a heavy quantum particle, called the tracer particle, coupled to an ideal gas of light Bose particles, the ratio of masses of the tracer particle and a gas particle being proportional to the gas density. All particles have non-relativistic kinematics. The tracer particle is driven by an external potential and couples to the gas particles through a pair potential. We compare the quantum dynamics of this system to an effective dynamics given by a Newtonian equation of motion for the tracer particle coupled to a classical wave equation for the Bose gas. We quantify the closeness of these two dynamics as the mean-field limit is approached (gas density ${\to \infty}$ ). Our estimates allow us to interchange the thermodynamic with the mean-field limit.     

Gross-Pitaevskii Theory of the Rotating Bose Gas

 We study the Gross-Pitaevskii functional for a rotating two-dimensional Bose gas in a trap. We prove that there is a breaking of the rotational symmetry in the ground state; more precisely, for any value of the angular velocity and for large enough values of the interaction strength, the ground state of the functional is not an eigenfunction of the angular momentum. This has interesting consequences on the Bose gas with spin; in particular, the ground state energy depends non-trivially on the number of spin components, and the different components do not have the same wave function. For the special case of a harmonic trap potential, we give explicit upper and lower bounds on the critical coupling constant for symmetry breaking. Received: 1 December 2001 / Accepted: 19 April 2002 Published online: 6 August 2002     

Boundary Problems for the Quantum Bose Gas

A relaxation kinetic equation that describes the behavior of a Bose gas is derived. The Kramers half-space problem on isothermal slip is treated. An analytical solution and the number-of-particle distribution function for particles flying toward a wall, in explicit form, are obtained. The dependence of the slip velocity on the parameter that is the ratio of the chemical potential to the product of Boltzmann's constant by the absolute temperature is analyzed. The influence of the quantum effects on the isothermal slip coefficient is evaluated for He⁴.     

Signature of Critical Point in Momentum Profile of Trapped Ultracold Bose Gases

We present a new method to identify the critical point for the Bose-Einstein condensation(BEC) of a trapped Bose gas.We calculate the momentum distribution of an interacting Bose gas near the critical temperature,and find that it deviates significantly from the Gaussian profile as the temperature approaches the critical point.More importantly,the standard deviation between the calculated momentum spectrum and the Gaussian profile at the same temperature shows a turning point at the critical point,which can be used to determine the critical temperature.These predictions are aiso confirmed by our BEC experiment for magnetically trapped ~(87)Rb gases.     

Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas

We measure the zero-temperature equation of state of a homogeneous Bose gas of (7)Li atoms by analyzing the in situ density distributions of trapped samples. For increasing repulsive interactions our data show a clear departure from mean-field theory and provide a quantitative test of the many-body corrections first predicted in 1957 by Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957).]. We further probe the dynamic response of the Bose gas to a varying interaction strength and compare it to simple theoretical models. We deduce a lower bound for the value of the universal constant ξ > 0.44(8) that would characterize the universal Bose gas at the unitary limit.     

Ground State Energy of the Two-Component Charged Bose Gas

We continue the study of the two-component charged Bose gas initiated by Dyson in 1967. He showed that the ground state energy for N particles is at least as negative as –CN^7/5 for large N and this power law was verified by a lower bound found by Conlon, Lieb and Yau in 1988. Dyson conjectured that the exact constant C was given by a mean-field minimization problem that used, as input, Foldys calculation (using Bogolubovs 1947 formalism) for the one-component gas. Earlier we showed that Foldys calculation is exact insofar as a lower bound of his form was obtained. In this paper we do the same thing for Dysons conjecture. The two-component case is considerably more difficult because the gas is very non-homogeneous in its ground state.Dedicated to Freeman J. Dyson on the occasion of his 80th birthday©2003 by the authors. This article may be reproduced in its entirety for non-commercial purposes.Work partially supported by NSF grant DMS-0111298, by EU grant HPRN-CT-2002-00277, by MaPhySto – A Network in Mathematical Physics and Stochastics, funded by The Danish National Research Foundation, and by grants from the Danish research council.Work partially supported by U.S. National Science Foundation grant PHY01 39984-A01.     

Avalanche Dynamics in Quenched Random Directed Sandpile Models

We numerically investigate the quenched random directed sandpile models which are local, conservative and Abelian. A local flow balance between the outflow of grains during a single toppling at a site and the total number of grains flowing into the same site plays an important role when all the nearest-neighbouring sites of the above-mentioned site topple for once. The quenched model has the same critical exponents with the Abelian deterministic directed sandpile model when the local flow balance exists, otherwise the critical exponents of this quenched model and the annealed Abelian random directed sandpile model are the same. These results indicate that the presence or absence of this local flow balance determines the universality class of the Abelian directed sandpile model.     

Ground State Energy of the One-Component Charged Bose Gas

The model considered here is the “jellium” model in which there is a uniform, fixed background with charge density −eρ in a large volume V and in which N=ρV particles of electric charge +e and mass m move – the whole system being neutral. In 1961 Foldy used Bogolubov's 1947 method to investigate the ground state energy of this system for bosonic particles in the large ρ limit. He found that the energy per particle is −0.402 in this limit, where . Here we prove that this formula is correct, thereby validating, for the first time, at least one aspect of Bogolubov's pairing theory of the Bose gas. Received: 23 August 2000 / Accepted: 5 October 2000     

Depinning Dynamics of Fluid Monolayer on a Quenched Substrate

Langevin simulations are preformed on the depinning dynamics of fluid monolayer on a quenched substrate. With increase in the strength of the substrate, we find for the first time a crossover from elastic crystal to smectic flows as well as a crossover from smectic to plastic flows above the depinning. A power-law scaling relationship can be derived between the drift velocity and the driving force for both the elastic crystal and smectic flows, but fails to be obtained for the plastic flow. The power-law exponents are found to be no larger than 1 for the elastic crystal flow and larger than 1 for the smeetic flow. The critical driving force and the averaged intensity of Bragg peaks remain invariant basically in the regime of smectic flow. A sudden increase in the critical driving force is observed within the crossover from the smeetic to plastic flows, and the averaged intensity of Bragg peaks shows sudden decreases within the crossovers both from the elastic crystal to smectic flows and from the smectic to plastic flows. The results are helpful for understanding the slip dynamics of fluids on a molecular level.     

Depinning Dynamics of Fluid Monolayer on a Quenched Substrate

Langevin simulations are preformed on the depinning dynamics of fluidmonolayer on a quenched substrate. With increase in the strength of thesubstrate, we find for the first time a crossover from elastic crystal tosmectic flows as well as a crossover from smectic to plastic flows above thedepinning. A power-law scaling relationship can be derived between the driftvelocity and the driving force for both the elastic crystal and smecticflows, but fails to be obtained for the plastic flow. The power-lawexponents are found to be no larger than 1 for the elastic crystal flow andlarger than 1 for the smectic flow. The critical driving force and theaveraged intensity of Bragg peaks remain invariant basically in the regimeof smectic flow. A sudden increase in the critical driving force is observedwithin the crossover from the smectic to plastic flows, and the averagedintensity of Bragg peaks shows sudden decreases within the crossovers bothfrom the elastic crystal to smectic flows and from the smectic to plastic flows.The results are helpful for understanding the slip dynamics of fluids on a molecular level.     

Fluctuations in the Bose Gas with Attractive Boundary Conditions

In this paper limiting distribution functions of field and density fluctuations are explicitly and rigorously computed for the different phases of the Bose gas. Several Gaussian and non-Gaussian distribution functions are obtained and the dependence on boundary conditions is explicitly derived. The model under consideration is the free Bose gas subjected to attractive boundary conditions, such boundary conditions yield a gap in the spectrum. The presence of a spectral gap and the method of the coupled thermodynamic limits are the new aspects of this work, leading to new scaling exponents and new fluctuation distribution functions.     

On Condensations in the Bogoliubov Weakly Imperfect Bose Gas

We show that condensation in the Bogoliubov weakly imperfect Bose gas (WIBG) may appear in two stages. If interaction is such that the pressure of the WIBG does not coincide with the pressure of the perfect Bose gas (PBG), then the WIBG may manifest two kinds of condensations: nonconventional Bose condensation in zero mode, due to the interaction (the first stage), and conventional (generalized) Bose–Einstein condensation in modes next to the zero mode due to the particle density saturation (the second stage). Otherwise the WIBG manifests only the latter kind of condensation.     

Isobaric Heat Capacity of an Ideal Bose Gas

It is demonstrated that the heat capacity of an ideal Bose gas at constant pressure increases infinitely when its temperature approaches the Bose condensation temperature from above and is infinite for the phase with a Bose condensate.     

Dynamics of Non- interacting System with Long-Range Correlated Quenched Impurities

The theoretic renormalization group approach is applied to the study of the critical behavior of non-interacting system with long-range correlated quenched impurities, which has a power-like correlations r-(d-ρ). Totwo-loop order, the asymptotic scaling laws and the critical exponents are studied in the frame of a double (ε, ρ)expansion with ρ of order ε = 4 - d. In d ＜ 4, it is argued that the initial slip exponent θ = 0 together with the dynamicexponent z ＜ 2 is exact in this kind of random system.     

Thermodynamics of Systems of Non-Equilibrium Momentum Distribution

There exist physical processes (e. g. in energetic heavy ion collisions), where the time scale of the process is not sufficiently long to reach complete thermal equilibrium. Here we show a way to extend the thermodynamical formalism for unequilibrium situations. The results are compatible with continuum mechanics.