Bose-Einstein condensation of interacting gases (original) (raw)
Related papers
Bose-Einstein condensation in interacting gases
The European Physical Journal B, 1999
We study the occurrence of a Bose-Einstein transition in a dilute gas with repulsive interactions, starting from temperatures above the transition temperature. The formalism, based on the use of Ursell operators, allows us to evaluate the one-particle density operator with more flexibility than in mean-field theories, since it does not necessarily coincide with that of an ideal gas with adjustable parameters (chemical potential, etc.). In a first step, a simple approximation is used (Ursell-Dyson approximation), which allow us to recover results which are similar to those of the usual mean-field theories. In a second step, a more precise treatment of the correlations and velocity dependence of the populations in the system is elaborated. This introduces new physical effects, such as a marked change of the velocity profile just above the transition: low velocities are more populated than in an ideal gas. A consequence of this distortion is an increase of the critical temperature (at constant density) of the Bose gas, in agreement with those of recent path integral Monte-Carlo calculations for hard spheres.
Bose–Einstein transition temperature in a dilute repulsive gas
Comptes Rendus Physique, 2004
We discuss certain specific features of the calculation of the critical temperature of a dilute repulsive Bose gas. Interactions modify the critical temperature in two different ways. First, for gases in traps, temperature shifts are introduced by a change of the density profile, arising itself from a modification of the equation of state of the gas (reduced compressibility); these shifts can be calculated simply within mean field theory. Second, even in the absence of a trapping potential (homogeneous gas in a box), temperature shifts are introduced by the interactions; they arise from the correlations introduced in the gas, and thus lie inherently beyond mean field theory-in fact, their evaluation requires more elaborate, non-perturbative, calculations. One illustration of this non-perturbative character is provided by the solution of self-consistent equations, which relate together non-linearly the various energy shifts of the single particle levels k. These equations predict that repulsive interactions shift the critical temperature (at constant density) by an amount which is positive, and simply proportional to the scattering length a; nevertheless, the numerical coefficient is difficult to compute. Physically, the increase of the temperature can be interpreted in terms of the reduced density fluctuations introduced by the repulsive interactions, which facilitate the propagation of large exchange cycles across the sample.
Bose-Einstein Condensation of An Ideal Gas
An ideal gas consisting of non-interacting Bose particles is a fictitious system since every realistic Bose gas shows some level of particle-particle interaction. Nevertheless, such a mathematical model provides the simplest example for the realization of Bose-Einstein condensation. This simple model, first studied by A. Einstein [1], correctly describes important basic properties of actual non-ideal (interacting) Bose gas. In particular, such basic concepts as BEC critical temperature T c (or critical particle density n c ), condensate fraction N 0 /N and the dimensionality issue will be obtained.
A Nonextensive Approach to Bose-Einstein Condensation of Trapped Interacting Boson Gas
Journal of Low Temperature Physics, 2008
In the Bose-Einstein condensation of interacting atoms or molecules such as 87Rb, 23Na and 7Li, the theoretical understanding of the transition temperature is not always obvious due to the interactions or zero point energy which cannot be exactly taken into account. The S-wave collision model fails sometimes to account for the condensation temperatures. In this work, we look at the problem within the nonextensive statistics which is considered as a possible theory describing interacting systems. The generalized energy U q and the particle number N q of boson gas are given in terms of the nonextensive parameter q. q>1 (q<1) implies repulsive (attractive) interaction with respect to the perfect gas. The generalized condensation temperature T cq is derived versus T c given by the perfect gas theory. Thanks to the observed condensation temperatures, we find q≈0.1 for 87Rb atomic gas, q≈0.95 for 7Li and q≈0.62 for 23Na. It is concluded that the effective interactions are essentially attractive for the three considered atoms, which is consistent with the observed temperatures higher than those predicted by the conventional theory.
We present a microscopic theory of the second-orderphase transition in an interacting Bose gas that allows one to describe formation of an ordered condensate phase from a disordered phase across an entire critical region continuously. We derive the exact fundamental equations for a condensate wave function and the Green’s functions, which are valid both inside and outside the critical region. They are reduced to the usual Gross–Pitaevskii and Beliaev–Popov equations in a low-temperature limit outside the critical region. The theory is readily extendable to other phase transitions, in particular, in the physics of condensed matter and quantum fields.
Condensate Statistics in Interacting and Ideal Dilute Bose Gases
Physical Review Letters, 2000
We obtain analytical formulas for the statistics, in particular, for the characteristic function and all cumulants, of the Bose-Einstein condensate in dilute weakly interacting and ideal equilibrium gases in the canonical ensemble via the particle-number-conserving operator formalism of Girardeau and Arnowitt. We prove that the ground-state occupation statistics is not Gaussian even in the thermodynamic limit. We calculate the effect of Bogoliubov coupling on suppression of ground-state occupation fluctuations and show that they are governed by a pair-correlation, squeezing mechanism. PACS numbers: 03.75.Fi, 05.30.Jp Two interesting recent papers have addressed the question of condensate fluctuations in the interacting Bose gas. Recently we have been working on the canonicalensemble approach to the condensation of N bosons in a trap using a nonequilibrium (laserlike) analysis [3,4] on the one hand, and a particle-number-conserving operator formalism [5-7] on the other.
The transition temperature of the dilute interacting Bose gas for N internal states
Europhysics Letters (EPL), 2000
We calculate explicitly the variation δT c of the Bose-Einstein condensation temperature T c induced by weak repulsive two-body interactions to leading order in the interaction strength. As shown earlier by general arguments, δT c /T c is linear in the dimensionless product an 1/3 to leading order, where n is the density and a the scattering length. This result is non-perturbative, and a direct perturbative calculation of the amplitude is impossible due to infrared divergences familiar from the study of the superfluid helium lambda transition. Therefore we introduce here another standard expansion scheme, generalizing the initial model which depends on one complex field to one depending on N real fields, and calculating the temperature shift at leading order for large N. The result is explicit and finite. The reliability of the result depends on the relevance of the large N expansion to the situation N = 2, which can in principle be checked by systematic higher order calculations. The large N result agrees remarkably well with recent numerical simulations.
Finite-Temperature Bose-Einstein Condensation in Interacting Boson System
Ukrainian Journal of Physics
Thermodynamical properties of an interacting boson system at finite temperatures and zero chemical potential are studied within the framework of the Skyrme-like mean-field toy model. It is assumed that the mean field contains both attractive and repulsive terms. Self-consistency relations between the mean field and thermodynamic functions are derived. It is shown that, for sufficiently strong attractive interactions, this system develops a first-order phase transition via the formation of a Bose condensate. An interesting prediction of the model is that the condensed phase is characterized by a constant total density of particles. It is shown that the energy density exhibits a jump at the critical temperature.