Fluctuations in the Bose Gas with Attractive Boundary Conditions.pdf

a r X i v : m a t h - p h / 0 1 1 1 0 0 9 v 2 3 0 A u g 2 0 0 2 Preprint-KUL-TF-2001/22 Fluctuations in the Bose Gas with Attractive Boundary Conditions J. Lauwers1, A. Verbeure2 Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium November 2001 Abstract In this paper limiting distribution functions of field and density fluctuations are ex- plicitly and rigorously computed for the different phases of the Bose gas. Several Gaussian and non-Gaussian distribution functions are obtained and the dependence on boundary conditions is explicitly derived. The model under consideration is the free Bose gas subjected to attractive boundary conditions, such boundary conditions yield a gap in the spectrum. The presence of a spectral gap and the method of the coupled thermodynamic limits are the new aspects of this work, leading to new scal- ing exponents and new fluctuation distribution functions. Keywords Quantum Fluctuations, Distribution Functions, Bose-Einstein Condensation, Bound- ary Conditions. 1Bursaal KUL FLOF-10408 1Email: joris.lauwers@fys.kuleuven.ac.be 2Email: andre.verbeure@fys.kuleuven.ac.be 1 Introduction Normal and critical fluctuations in the ideal Bose gas with Dirichlet or periodic boundary conditions are explicitly studied at different levels of rigour in [1–10]. Condensation occurs only at dimensions ν ≥ 3. It turns out that the behaviour of the energy gap of the finite volume spectrum as a function of the volume is determining for the degree of abnormality of the fluctuations in the condensate regime. Typical is that the limit spectrum has no energy gap. The ideal Bose gas with attractive boundary conditions [11, 12] on the other hand is quite different in nature. Condensation is possible in all space dimensions, and the spectrum has a finite energy gap in the thermodynamic limit. In this paper we analyse the nature of field and density fluctuations for the Bose gas with attractive boundary c