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Journal of Mathematical Physics

Bose-Einstein condensation in the presence of impurities

Mark Kac, J. M. Luttinger

Published Nov 1, 1973• 47 citations
Publisher

Abstract

It is shown that, in three dimensions, a gas of noninteracting bosons in the presence of purely repulsive impurity centers undergoes a ``Bose-Einstein'' phase transition at sufficiently low temperature. In the course of the proof it is also shown that, as the size of the system approaches infinity, the lowest energy state approaches zero with probability one.

DOI
10.1063/1.1666234

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It is shown that, in three dimensions, a gas of noninteracting bosons in the presence of purely repulsive impurity centers undergoes a ``Bose-Einstein'' phase transition at sufficiently low temperature. In the course of the proof it is also shown that, as the size of the system approaches infinity, the lowest energy state approaches zero with probability one.
We study the thermodynamics of a one-dimensional system of noninteracting bosons in the presence of random fixed finite-range repulsive potentials. It is shown that a Bose---Einstein condensation into the lowest state occurs. The critical temperature increases with the number of impurities. The specific heat exhibits a discontinuity and is infinitely differentiable from both sides of the critical point. For a suitably defined pressure the isothermals exhibit a flat portion below the "transition length."
The behavior of an ideal D-dimensional boson gas in the presence of a uniform gravitational field is analyzed. It is explicitly shown that, contrary to a long-standing belief, the three-dimensional gas does not undergo Bose-Einstein condensation at finite temperature. On the other hand, Bose-Einstein condensation occurs at $T\ensuremath{\ne}0$ for D=1,2,3D=1,2,3D=1,2,3 if there is a pointlike impurity at the bottom of the vessel containing the gas.
The thermodynamic properties of an ideal gas of bosons moving freely within layered planes and having a finite amplitude for jumping between neighbouring planes, are derived. As long as this coupling is non-zero, the system becomes effectively two-dimensional at sufficiently high temperatures. At a correspondingly low temperature it undergoes a Bose-Einstein transition into a condensed phase as in three dimensions. Below the critical temperature the specific heat is a universal function of only one dimensionless system variable. The phase transition becomes weaker as the coupling between the planes decreases and eventually disappears when the planes decouple.
The exact density of states for one particle in the field of an attractive $\ensuremath{\delta}$ impurity center is obtained from the system Green's function. The impurity is shown to introduce a bound state in the free-particle energy spectrum without any further modification. This result is then used to study the effects of the impurity on the thermodynamical properties of a gas of noninteracting bosons. The "Bose-Einstein phase transition" is not destroyed by the impurity, but a true condensation in position space occurs due to the existence of the bound state. The captured bosons form a cloud of well-localized particles modifying the macroscopic behavior of the gas for temperatures below a transition temperature Tc{T}_{c}Tc​. The constant-volume specific heat, the entropy, and the pressure are evaluated as functions of the temperature and of the energy of the bound state, Eb{E}_{b}Eb​. The transition temperature and the discontinuity of the specific heat at Tc{T}_{c}Tc​ are also obtained as a function of Eb{E}_{b}Eb​.
The thermodynamic behavior of the one-dimensional Bose-gas–attractive-δ impurity system is studied in this paper. The system is shown to undergo the Bose–Einstein condensation and the cause of the phase transition is attributed to the bound state introduced by the impurity in the free-particle energy spectrum. The condensed phase is composed of particles captured by the impurity, forming a drop of particles well localized in space. This gives to the Bose–Einstein condensation in this system the appearance of the ordinary vapor–liquid phase transition. The expression for the pressure is obtained and is plotted as a function of the temperature and the system length. The shapes of the isothermal curves reinforce the interpretation of a vapor–liquid transition. The evaluation of the heat capacity at constant length shows the existence of a finite discontinuity at the transition temperature.
We study condensation of trapped bosons in the limit when the number of particles tends to infinity. For the noninteracting gas we prove that there is no phase transition in any dimension, but in any dimension at any temperature the system is 100% condensated into the one-particle ground state. In the case of an interacting gas we show that for a family of suitably scaled pair interactions, the Gross-Pitaevskii scaling included, a less-than-100% condensation into a one-particle eigenstate, which may depend on the interaction strength, persists at all temperatures.