The Theory of a Perfect Bose Gas at a Temperature Different from Zero

Victor Nikolaevich Popov, Lyudvig Dmitrievich Faddeev

Robert H. Kraichnan

The nature of the Einstein-Bose condensation in an imperfect boson gas is discussed. It is argued that no matter how low the temperature, provided it is not zero, macroscopic second-sound waves should be weakly excited as thermal fluctuations about equilibrium. This implies that the single-particle momentum distribution must be smeared so that the state of zero momentum cannot contain a finite fraction of the particles. A microscopic mechanism which independently leads to a momentum spread is discussed. It is suggested that the actual Einstein-Bose condensation of an imperfect gas takes place in energy rather than momentum but that even in terms of this variable it becomes perfectly sharp only at zero temperature. This hypothesis is formulated analytically in terms of correlation functions of the second-quantized boson field.

Yu. V. Sanochkin

Juan José Valencia Acevedo, M. Atala

Abstract We report the effects on the thermodynamic properties of a 3D Bose gas caused by a temperature dependent energy gap ∆ (T) at the lower edge of the energy spectrum of the particles constituting the Bose gas which behaves like an ideal Bose gas when the gap is removed. Explicit formulae are given for the critical temperature, the condensate fraction, the internal energy and the isochoric specific heat, which are calculated for three different gaps that abruptly go to zero at temperature TB, as well as for the damped counterparts whose drop to zero we have smoothed. In particular, for the undamped BCS (Bardeen, Cooper and Schrieffer) gap it is observed that the Bose-Einstein condensation (BEC) critical temperature Tc is equal to that of the ideal Bose gas T0, for all TB ≤ T0; surprisingly, the condensate fraction presents two different filling rates of the ground state at Tc = T0 and at TB < T0 while the specific heat shows a finite jump at Tc as well as a divergence at TB. Three-dimensional infinite Bose gas results are recovered when the temperature independent gap is either a constant or equal to zero. PACS numbers: 03.75.Fi; 05.30.Jp; 67.40.Kh

C. J. Pethick, H. Smith

R. H. Fowler, Hilary D. Jones

In a recent letter to Nature F. London(3) has called attention to a discontinuity in the derivative dCV/dT of the specific heat with respect to temperature which arises in a perfect Einstein-Bose gas at low temperatures. When discussing the properties of a perfect gas satisfying the Bose statistics, Einstein(1) remarked that at low temperatures something resembling a condensed phase should appear. The temperature at which this condensation should begin is given by the equation where n is the number of particles of mass m per unit volume and the other symbols have their usual significance.