The Modified Group Expansions for Construction of Solutions to the BBGKY Hierarchy

A solution to the BBGKY hierarchy for nonequilibrium distribution functions is obtained within modified boundary conditions. The boundary conditions take into account explicitly both the nonequilibrium one-particle distribution function as well as local conservation laws. As a result, modified group expansions are proposed. On the basis of these expansions, a generalized kinetic equation for hard spheres and a generalized Bogolubov–Lenard–Balescu kinetic equation for a dense electron gas are derived within the polarization approximation.     

Stability of coherent states of a Bose gas of two-level atoms in resonant laser field at zero temperature

It is shown that a two-level atom Bose gas in a strong resonant laser field at zero temperature is a mixture of two condensates with a certain density ratio. The stability criteria for stationary states of such a system relative to the increase in the amplitudes of quasi-Bogolyubov elementary excitations of the Bose gas are formulated. It is shown that, in addition to conventional acoustic mode, a mode gap exists with a gap proportional to the laser field amplitude. Under certain conditions, deviations from ideality of the gas may lead to instability and decomposition of the condensate.     

The effect of repulsive interactions on Bose-Einstein condensation

One-dimensional Bose gases that interact via a repulsive two-body interaction and show Bose-Einstein condensation at the free level are studied. It is shown that the introduction of this interaction, however small, destroys the condensate. It is also shown that the free energy of an interacting Bose gas does not depend on the boundary conditions(including attractive boundary conditions) in the van der Waals limit.     

On the Bose gas with local mean-field interaction

A Bose gas model is considered where the interaction term is proportional to the integral over the square of the local particle density. This model exhibits a phase transition with the same critical behavior as the usual mean-field (imperfect) Bose gas, but only generalized condensation may occur, due to boundary conditions.     

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.     

Renormalization of boundary conditions for distribution functions of quasiparticles obeying quantum statistics at interfaces between crystalline grains

Boundary conditions for distribution functions of quasiparticles scattered by an interface between two crystalline grains are presented. In contradistinction to former formulations where the Maxwell-Boltzmann statistics was considered, the present boundary conditions take into account the quantum (Fermi-Dirac or Bose-Einstein) statistics of the quasiparticles. Provided that small deviations from the thermodynamical equilibrium only are present, the boundary conditions are linearized, and then their “renormalization” is investigated in case of the elastic scattering. The final results of the renormalization, which are obtained for a simplified model of an interface, suggest that the portion of the Fermi (Bose)-quasiparticles reflected or transmitted specularly is decreased (increased) in comparison with the case of quasiparticles obeying the Maxwell-Boltzmann statistics.     

Renormalization group analysis of a confined, interacting Bose gas

The renormalization group is not only a powerful method for describing universal properties of phase transitions, but it is also useful for evaluating non-universal thermodynamic properties beyond mean-field theory. In this contribution we concentrate on these latter aspects of the renormalization group approach. We introduce its main underlying ideas in the familiar context of the ideal Bose gas and then apply them to the case of an interacting, confined Bose gas within the framework of the random phase approximation. We model confinement by periodic boundary conditions and demonstrate how confinement modifies the flow equations of the renormalization group, thus changing the thermodynamic properties of the gas. Received: 20 July 2001 / Revised version: 20 August 2001 / Published online: 23 November 2001     

Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation

Rarefied gas flow behavior is usually described by the Boltzmann equation, the Navier-Stokes system being valid when the gas is less rarefied. Slip boundary conditions for the Navier-Stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level (Boltzmann equation). These slip conditions are explicitly written in terms of asymptotic behavior of some linear half-space problems. The validity of this analysis is established in the simple case of the Couette flow, for which it is proved that the right boundary conditions are obtained.     

On a class of equilibrium states under the Kubo-Martin-Schwinger condition

We study equilibrium states of a quantum Bose gas using Kubo-Martin-Schwinger boundary conditions, for a special class of time evolutions, namely the quasi-free evolutions. Under suitable restrictions, in particular positivity of the elementary excitation spectrum, we are able to describe the states fulfilling the Kubo-Martin-Schwinger conditions. In contrast to the Fermi case the solution is, in general, not unique; this is related to a possible Bose condensation.Attaché de Recherche, C.N.R.S.This work is a part of a «Thèse de Doctorat d'Etat» presented to the «Faculté des Sciences de Marseille», under the number A.O. 3680.     

Photoelektrische Registrierung der Linienprofile thermisch angeregter He I-Linien

The self-energies defined in a preceding paper are calculated explicitly for a dilute Bose gas with weak repulsive interaction. The transport equations are written down and the coupled system of kinetic equations are solved in order to calculate the propagation of normal and second sound.     

Painlevé Transcendent Evaluations of Finite System Density Matrices for 1d Impenetrable Bosons

 The recent experimental realisation of a one-dimensional Bose gas of ultra cold alkali atoms has renewed attention on the theoretical properties of the impenetrable Bose gas. Of primary concern is the ground state occupation of effective single particle states in the finite system, and thus the tendency for Bose-Einstein condensation. This requires the computation of the density matrix. For the impenetrable Bose gas on a circle we evaluate the density matrix in terms of a particular Painlevé VI transcendent in Σ-form, and furthermore show that the density matrix satisfies a recurrence relation in the number of particles. For the impenetrable Bose gas in a harmonic trap, and with Dirichlet or Neumann boundary conditions, we give a determinant form for the density matrix, a form as an average over the eigenvalues of an ensemble of random matrices, and in special cases an evaluation in terms of a transcendent related to Painlevé V and VI. We discuss how our results can be used to compute the ground state occupations. Received: 24 July 2002 / Accepted: 26 January 2003 Published online: 13 May 2003 Communicated by L. Takhtajan     

Families of bose rays in quantum optics

Having known classical wave optics and wave mechanics, can we reverse Schrödinger's path and extend the concept of families of rays of light to provide a new exact rendering of quantum optics including the Bose nature of photons? This question is answered in the affirmative, and the implications of the Bose symmetry for certain nonlocal correlations of the many-ray distribution functions are worked out. The similarities and the differences between classical and quantum wave optics are brought out. The ray-ray Bose correlation is analyzed. The generating functional for the many-ray distribution functions is formulated; and the notion of paraxial illumination for quantum optics is made precise.     

The condensed state of the imperfect Bose gas

For the imperfect Bose gas, a rigorous proof is given for condensation in and only in the ground state, and the limit Gibbs state is explicitly constructed.     

On the nonequilibrium statistical mechanics of a binary mixture. I. The distribution functions

In this paper, we obtain the generalization of the BBGKY hierarchy for a binary mixture of chemically neutral particles. Using modified boundary conditions different from the ones proposed by Bogoliubov, we solve the hierarchy, and obtain explicitly the set of two-particle distribution functions for the several species of the mixture, up to first order in the density.     

Gentle perturbations of the free Bose gas. I

It is demonstrated that the thermal structure of the noncritical free Bose gas is completely described by certain periodic generalized Gaussian stochastic process or equivalently by a certain periodic generalized Gaussian random field. Elementary properties of this Gaussian stochastic thermal structure are established. Gentle perturbations of several types of the free thermal stochastic structure are studied. In particular, new models of non-Gaussian thermal structures are constructed and a new functional integral representation of the corresponding Euclidean-time Green functions is obtained rigorously.     

Distribution of eigenvalues for the modular group

The two-point correlation functions of energy levels for free motion on the modular domain, both with periodic and Dirichlet boundary conditions, are explicitly computed using a generalization of the Hardy-Littlewood method. It is shown that in the limit of small separations they show an uncorrelated behaviour and agree with the Poisson distribution but they have prominent number-theoretical oscillations at larger scale. The results agree well with numerical simulations.     

Violation of the Porod law in a freely cooling granular gas in one dimension

We study a model of freely cooling inelastic granular gas in one dimension, with a restitution coefficient which approaches the elastic limit below a relative velocity scale delta. While at early times (tdelta;{-1}) it exhibits a new fluctuation-dominated phase ordering state. We find distinct scaling behavior for the (i) density distribution function, (ii) occupied and empty gap distribution functions, (iii) the density structure function, and (iv) the velocity structure function, as compared to the completely inelastic sticky gas. The spatial structure functions (iii) and (iv) violate the Porod law. Within a mean-field approximation, the exponents describing the structure functions are related to those describing the spatial gap distribution functions.     

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.     

Relativistic charged Bose gas

The excitation spectrum of a relativistic spin-zero charged Bose gas is obtained in a dielectric response formulation. Relativity introduces a dip in the spectrum and the consequences of this dip for the thermodynamic functions are discussed.     

三维简谐势阱中玻色-爱因斯坦凝聚的边界效应

在定义特征长度的基础上,应用Euler–MacLaurin公式,研究了理想玻色气体在三维简谐势阱中玻色-爱因斯坦凝聚的边界效应.结果表明:粒子的凝聚分数由于有限尺度和有限粒子数效应而减小,修正的凝聚分数和凝聚温度由于边界效应存在一个极大值,选择优化的最佳势阱参数,可以有效提高凝聚分数和凝聚温度;热容量的跃变存在边界效应和粒子数效应,选择合理的势阱参数时,热容量的跃变存在一个极小值.导出了简谐势阱中有限理想玻色气体的状态方程,揭示了压强的各向异性(或各向同性)取决于简谐势频率的各向异性(或各向同性).