On the Discretization of the Coupled Integrable Dispersionless Equations

We study the integrable discretization of the coupled integrable dispersionless equations. Two semi-discrete version and one full-discrete version of the system are given via Hirota's bilinear method. Soliton solutions for the derived discrete systems are also presented.     

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 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.     

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.     

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.     

Analytical Number Theory and the Energy of Transition of Bose Gas to Fermi gas. Critical Lines as Boundaries of Noninteracting Gas (an Analog of the Bose Gas in Classical Thermodynamics)

It is proved that certain distributions in the analytic number theory coincide with the Bose–Einstein distribution. The transition of the boson branch of the decomposition of an integer (with repeated summands) into the fermion branch (without repeated summands) is described in detail near a small value of activity. Analytic formulas for the energy of transition of Bose gas to Fermi gas are obtained in the three-dimensional case and the nine-dimensional case (diatomic molecule). The radius of the Bose gas “jump” in the transition to the Fermi gas is calculated. The relationship between the constructed concepts and thermodynamics is described based on the obtained experimental values of gas characteristics on critical lines.     

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.     

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     

The Bose Gas and Asymmetric Simple Exclusion Process on the Half-Line

In this paper we find explicit formulas for: (1) Green’s function for a system of one-dimensional bosons interacting via a delta-function potential with particles confined to the positive half-line; and (2) the transition probability for the one-dimensional asymmetric simple exclusion process (ASEP) with particles confined to the nonnegative integers. These are both for systems with a finite number of particles. The formulas are analogous to ones obtained earlier for the Bose gas and ASEP on the line and integers, respectively. We use coordinate Bethe Ansatz appropriately modified to account for confinement of the particles to the half-line. As in the earlier work, the proof for the ASEP is less straightforward than for the Bose gas.     

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.     

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.     

An Integrable Discretization of the Rational {\mathfrak su(2)} Gaudin Model and Related Systems

The first part of the present paper is devoted to a systematic construction of continuous-time finite-dimensional integrable systems arising from the rational Gaudin model through certain contraction procedures. In the second part, we derive an explicit integrable Poisson map discretizing a particular Hamiltonian flow of the rational Gaudin model. Then, the contraction procedures enable us to construct explicit integrable discretizations of the continuous systems derived in the first part of the paper.     

Ground-State Correlation Functions for an Impenetrable Bose Gas with Neumann or Dirichlet Boundary Conditions

We study density correlation functions for an impenetrable Bose gas in a finite box, with Neumann or Dirichlet boundary conditions in the ground state. We derive the Fredholm minor determinant formulas for the correlation functions. In the thermodynamic limit, we express the correlation functions in terms of solutions of nonlinear differential equations which were introduced by Jimbo, Miwa, Môri, and Sato as a generalization of the fifth Painlevé equations.     

Dynamical Correlation Functions for an Impenetrable Bose Gas with Neumann or Dirichlet Boundary Conditions

Abstract

We study the time and temperature dependent correlation functions for an impenetrable Bose gas with Neumann or Dirichlet boundary conditions ψ(x ₁, 0)ψ ^?(x ₂ , t) _±,T . We derive the Fredholm determinant formulae for the correlation functions, by means of the Bethe Ansatz. For the special case x ₁ = 0, we express correlation functions with Neumann boundary conditions ψ(0, 0)ψ ^?(x ₂ , t) _+,T , in terms of solutions of nonlinear partial differential equations which were introduced in [1] as a generalization of the nonlinear Schrödinger equations. We generalize the Fredholm minor determinant formulae of ground state correlation functions ψ(x ₁)ψ ^?(x ₂) _±,0 in [2], to the Fredholm determinant formulae for the time and temperature dependent correlation functions ψ(x ₁, 0)ψ ^?(x ₂ , t) _±,T , t ∈ R, T ≥ 0.     

Repulsive synchronization in an array of phase oscillators

We study the dynamics of a repulsively coupled array of phase oscillators. For an array of globally coupled identical oscillators, repulsive coupling results in a family of synchronized regimes characterized by zero mean field. If the number of oscillators is sufficiently large, phase locking among oscillators is destroyed, independently of the coupling strength, when the oscillators' natural frequencies are not the same. In locally coupled networks, however, phase locking occurs even for nonidentical oscillators when the coupling strength is sufficiently strong.     

Low-Density Spinor Bose Gas of Particles with Arbitrary Spin

JETP Letters - The properties of the ground state and the spectrum of elementary excitations of an ultracold rarefied three-dimensional Bose gas of particles with arbitrary nonzero spin are...