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Physical Review

Many-Body Problem in Quantum Statistical Mechanics. V. Degenerate Phase in Bose-Einstein Condensation

T.D. Lee, Chen Ning Yang

Published Feb 15, 1960• 59 citations
Publisher

Abstract

The formulation of the previous paper (paper IV) is extended so that it becomes applicable in an interacting system in the presence of a Bose-Einstein degeneracy. This extension is carried out by the introduction of an $x$-ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.

DOI
10.1103/physrev.117.897

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The formulation of the previous paper (paper IV) is extended so that it becomes applicable in an interacting system in the presence of a Bose-Einstein degeneracy. This extension is carried out by the introduction of an xxx-ensemble, which enables one to utilize an Ursell-type expansion even in the presence of a Bose-Einstein degeneracy. The variational principle of the previous paper is also extended. It is proved that in the presence of a Bose-Einstein degeneracy, the average occupation number of a single particle state with momentum p approaches infinity as p\ensuremath{\rightarrow}0. The method is applied to a dilute system of Bose hard spheres.
Starting from Rules AAA and BBB of a previous paper (I), it is shown that the grand partition function can be evaluated in terms of the statistical averages of the occupation number in momentum space. The final formulation is in terms of a simple variational principle. The procedure represents a concise and complete separation of the effect of the Bose-Einstein or Fermi-Dirac statistical character of the particles from the dynamical problem. In the case of Bose statistics, this formulation makes possible a systematic computation of all thermodynamic functions near the Bose-Einstein transition point in the gaseous phase. Applications to a system of hard spheres are discussed.
We study the properties of quantum gases which are dilute in terms of the interactions but degenerate; this may be the case if the thermal wavelength of the particles is much larger than the interatomic potential range. We assume that the potential contains a short range repulsive part, which creates strong correlations between particles that cannot be treated by ordinary perturbation theory, even if the region of configuration space where they are dominant is relatively small. Our method combines two basic ingredients: Ursell operators, which generalize the Ursell functions that are usually used to generate quantum cluster expansions, but are not in themselves symmetrized (the operators can be related to an auxiliary system of distinguishable particles); in a distinct step, an exact treatment of particle indistinguishability is given through the inclusion of exchange cycles of arbitrary length. For non-condensed boson systems, as well as for fermion systems, the method provides a compact expression of the partition function that generalizes the well-known Beth–Uhlenbeck formula to arbitrary degeneracy. For the study of the Bose–Einstein condensation, our treatment retains more information on the short range correlations than, for instance, the Hartree–Fock method. It also introduces a natural and explicit distinction between particles and quasiparticles; the latter may condense into a single one-particle state, which obeys a non-linear equation somewhat similar to the Gross–Pitaevskii equation.
The theory of a degenerate system of interacting bosons is developed using the xxx-ensemble formulation of quantum statistics, previously introduced by Lee and Yang. Particular attention is devoted to the self-energy structure of the graphs of the theory, and it is shown that the complete analysis of this structure involves an intimate mixing of the effects of particle statistics and particle interactions. Therefore, the effect of statistics, which causes Bose-Einstein condensation, becomes mixed with the dynamics in a real system. A prescription is given for the grand potential in which all self-energy effects are included in the line factors of master graphs. An expression for the pair-distribution function of a degenerate Bose system is also derived.
We study in more detail the properties of the generalized Beth Uhlenbeck formula obtained in a preceding article. This formula leads to a simple integral expression of the grand potential of the system, where the interaction potential appears only through the matrix elements of the second order Ursell operator U2U_{2}U2​. Our results remain valid for significant degree of degeneracy of the gas, but not when Bose Einstein (or BCS) condensation is reached, or even too close from this transition point. We apply them to the study of the thermodynamic properties of degenerate quantum gases: equation of state, magnetic susceptibility, effects of exchange between bound states and free particles, etc. We compare our predictions to those obtained within other approaches, especially the ``pseudo potential'' approximation, where the real potential is replaced by a potential with zero range (Dirac delta function). This comparison is conveniently made in terms of a temperature dependent quantity, the ``Ursell length'', which we define in the text. This length plays a role which is analogous to the scattering length for pseudopotentials, but it is temperature dependent and may include more physical effects than just binary collision effects; for instance at very low temperatures it may change sign or increase almost exponentially, an effect which is reminiscent of a precursor of the BCS pairing transition. As an illustration, numerical results for quantum hard spheres are given.
We apply the formalism of Ursell operators, introduced in a previous article, to the calculation of reduced density operators (one and two-body density operators) in a dilute quantum gas; the calculation is not a fugacity expansion and is therefore not limited to low degrees of degeneracy. We obtain quantum corrections for the one-particle density operator as a function of the second Ursell operator. For the two-body density operator, we examine how statistics and interactions combine their effects on the correlations between particles; in particular we discuss in detail how hard cores potentials affect the short range correlations, a non-perturbative effect.
We consider Bose-Einstein condensation of noninteracting homogeneous three-dimensional gas in canonical ensemble when both particle number NNN and total momentum P\mathbf{P}P of all particles are fixed. Using the saddle-point method, we derive the large-NNN analytical approximations for partition function, free energy, and statistical distributions of occupation numbers of different single-particle energy levels. At temperatures below the critical point of phase transition, we predict, in some ranges of P\mathbf{P}P, fragmentation of the condensate, when more than one single-particle level is macroscopically occupied. The occupation number distributions have approximately Gaussian shapes for the levels hosting the condensate, and exponential shapes for other, noncondensate levels. Our analysis demonstrates breaking of Galilean invariance of moving finite-temperature many-particle system in the presence of Bose-Einstein condensation and extends the theory of moving and rotating quantum systems to the finite-temperature large-NNN limit.