Book contents
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The non-interacting Bose gas
- 3 Atomic properties
- 4 Trapping and cooling of atoms
- 5 Interactions between atoms
- 6 Theory of the condensed state
- 7 Dynamics of the condensate
- 8 Microscopic theory of the Bose gas
- 9 Rotating condensates
- 10 Superfluidity
- 11 Trapped clouds at non-zero temperature
- 12 Mixtures and spinor condensates
- 13 Interference and correlations
- 14 Fermions
- Appendix. Fundamental constants and conversion factors
- Index
11 - Trapped clouds at non-zero temperature
Published online by Cambridge University Press: 06 July 2010
- Frontmatter
- Contents
- Preface
- 1 Introduction
- 2 The non-interacting Bose gas
- 3 Atomic properties
- 4 Trapping and cooling of atoms
- 5 Interactions between atoms
- 6 Theory of the condensed state
- 7 Dynamics of the condensate
- 8 Microscopic theory of the Bose gas
- 9 Rotating condensates
- 10 Superfluidity
- 11 Trapped clouds at non-zero temperature
- 12 Mixtures and spinor condensates
- 13 Interference and correlations
- 14 Fermions
- Appendix. Fundamental constants and conversion factors
- Index
Summary
In this chapter we consider selected topics in the theory of trapped gases at non-zero temperature when the effects of interactions are taken into account. The task is to extend the considerations of Chapters 8 and 10 to allow for the trapping potential. In Sec. 11.1 we begin by discussing energy scales, and then calculate the transition temperature and thermodynamic properties. We show that at temperatures of the order of Tc the effect of interactions on thermodynamic properties of clouds in a harmonic trap is determined by the dimensionless parameter N1/6a/ā. Here ā, which is defined in Eq. (6.24), is the geometric mean of the oscillator lengths for the three principal axes of the trap. Generally this quantity is small, and therefore under many circumstances the effects of interactions are small. At low temperatures, thermodynamic properties may be evaluated in terms of the spectrum of elementary excitations of the cloud in its ground state, which we considered in Secs. 7.2, 7.3, and 8.2. At higher temperatures it is necessary to take into account thermal depletion of the condensate, and useful approximations for thermodynamic functions may be obtained using the Hartree–Fock theory as a starting point.
The remainder of the chapter is devoted to non-equilibrium phenomena. As we have seen in Secs. 10.3–10.5, two ingredients in the description of collective modes and other non-equilibrium properties of uniform gases are the two-component nature of condensed Bose systems, and collisions between excitations. For atoms in traps a crucial new feature is the in homogeneity of the gas. This in itself would not create difficulties if collisions between excitations were sufficiently frequent that matter remained in thermodynamic equilibrium locally.
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- Information
- Bose–Einstein Condensation in Dilute Gases , pp. 289 - 319Publisher: Cambridge University PressPrint publication year: 2001