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Journal de physique

Ideal Bose Einstein condensation and disorder effects

P. Lacour-Gayet, G. Toulouse

Published Jan 1, 1974• 93 citations
Publisher

Abstract

Ideal Bose Einstein condensation is studied as a cooperative phase transition for arbitrary dimensionality. The correspondence with the n → ∞ limit of the n-vector model in the constant volume case and with the n = — 2 limit in the constant pressure case is discussed and precised. The influence of a special type of disorder (random sources and sinks) is studied ; a sharp phase transition occurs with new critical exponents which are calculated and shown to violate some scaling laws.

DOI
10.1051/jphys:01974003505042500

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Ideal Bose Einstein condensation is studied as a cooperative phase transition for arbitrary dimensionality. The correspondence with the n → ∞ limit of the n-vector model in the constant volume case and with the n = — 2 limit in the constant pressure case is discussed and precised. The influence of a special type of disorder (random sources and sinks) is studied ; a sharp phase transition occurs with new critical exponents which are calculated and shown to violate some scaling laws.
Classic and recent results for the critical behaviour of ideal Bose gas at constant volume and constant pressure and for various spatial dimensionalities d>0 are reviewed. New results about the critical properties in a close vicinity of the λ−λ-λ−point are presented.
Properties of Bose–Einstein condensates (BEC) are studied in the presence of a Gaussian random potential of arbitrary strength of disorder and with arbitrary correlation length. Using the stochastic mean field approximation, a system having arbitrary strength of interactions is treated at finite temperature. A special case, in which we assume that the second moment of the random potential is Lorentzian, is treated in more detail at zero temperature. The total density of particles consists of condensate density, glassy density and normal density components. Our results are in agreement with those of Huang and Meng (HM). The system is stable below the critical value of the disorder parameter and undergoes a first-order phase transition at the critical value. We find that the critical value of the disorder parameter changes with changing the correlation length of the random potential.
Ideal Bose-Einstein condensation (BEC) remains a paradigmatic example of a continuous phase transition and a cornerstone for understanding quantum degenerate bosonic matter. We demonstrate that critical behavior of the ideal Bose gas near the BEC phase transition falls into three distinct classes, determined exclusively by the low-energy scaling of the density of states. Depending on its scaling exponent, which is controlled by dimensionality and confinement, the transition displays either the usual algebraic divergences of thermodynamic susceptibilities, divergent behavior with marginal logarithmic corrections, or a more subtle form of criticality, where only the correlation length diverges. Our work provides a unified framework for criticality in noninteracting bosonic systems. This classification applies broadly to atomic, photonic, polaritonic, and magnonic condensates, where dimensionality, confinement, and spectral engineering can strongly reshape the density of states.
Bose–Einstein condensation (BEC) is discussed at the level of an advanced course of statistical thermodynamics, clarifying some formal and physical aspects that are usually not covered by the standard pedagogical literature. The non-conventional approach adopted starts by showing that the continuum limit, in certain cases, cancels out the crucial role of the bosonic ground level. If so, a correct treatment of the problem, including the ground level population N 0 by construction, leads to BEC in a straightforward way. For a density of states of the form G ( )∝ γ , the chemical potential µ is explicitly calculated as a function of the temperature T and of the number N of bosons, for various significant values of the positive exponent γ. In the thermodynamic limit, in which the boson number N diverges and BEC is a sharp process, the chemical potential µ is a singular function of T at the critical temperature T B , determined by an appropriate critical exponent. The condensate population N 0 is studied analytically and numerically as a function of the temperature, for various values of N and for different γ. This provides an accurate description of the way BEC approaches the character of a sharp phase transition. Some aspects of the real experiments on BEC, involving a finite number of bosons, are also illustrated.