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arXiv:0810.4001 · Journal of Physics A Mathematical and Theoretical

A scaling approach to the existence of long cycles in Casimir boxes

Mathieu Beau

Published May 22, 2009• 7 citations
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Abstract

We analyze the concept of generalized Bose–Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the off-diagonal-long-range-order (ODLRO), which are usually considered as an adequate way to describe standard BEC on the ground state for cubic boxes. First, we show that these three criteria are equivalent in this latter case. Then, based on a scaling approach, we revise the formulation of these concepts to prove that the classification of g-BEC into three types I, II, III, implies a hierarchy of long cycles (depending on their size scale) as well as a hierarchy of ODLRO which depends on the coherence length of the condensate.

DOI
10.1088/1751-8113/42/23/235204

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We analyze the concept of generalized Bose–Einstein condensation (g-BEC), known since 1982 for the perfect Bose gas (PBG) in Casimir (or anisotropic) boxes. Our aim is to establish a relation between this phenomenon and two concepts: the concept of long cycles and the off-diagonal-long-range-order (ODLRO), which are usually considered as an adequate way to describe standard BEC on the ground state for cubic boxes. First, we show that these three criteria are equivalent in this latter case. Then, based on a scaling approach, we revise the formulation of these concepts to prove that the classification of g-BEC into three types I, II, III, implies a hierarchy of long cycles (depending on their size scale) as well as a hierarchy of ODLRO which depends on the coherence length of the condensate.
It is shown for boson systems with periodic boundary conditions that the existence of generalized Bose condensation, of which the simple type typified by the ideal Bose gas is a special case, is equivalent to the existence of off-diagonal long-range order (ODLRO) in the single-particle density matrix ρ1 provided that the two noncomuting limits involved in the criterion for ODLRO are taken in the proper order: first size of system → ∞, then interparticle separation → ∞. It is shown by means of an example that certain assumptions concerning the behavior of the single-particle momentum distribution function involved in proving this equivalence are actually satisfied in thermal equilibrium for some dynamical models. The analysis is generalized to box-enclosure boundary conditions by an extension of an argument due to Schafroth, according to which box-enclosure conditions are equivalent to homogeneous Neumann conditions except for a thermodynamically negligible surface effect, provided that one second-quantizes with respect to Hartree-Fock orbitals rather than free-particle orbitals. A general criterion for generalized Bose condensation in terms of eigenvalues of ρ1 is proposed. On the basis of the behavior of soluble models it is conjectured that for a boson system in thermal equilibrium subject to arbitrary boundary conditions, the existence of such Bose condensation is equivalent to the existence of ODLRO in ρ1. The two-particle density matrix ρ2 is discussed briefly. By means of a simplified model it is shown that for a Bose system generalized condensation implies large eigenvalues of ρ2 and ODLRO of ρ2, just as for ρ1. It is pointed out that the question of the existence or nonexistence of generalized Bose condensation of fermion pairs ought to be investigated.
We consider the non-interacting Bose gas of N bosons in dimension d≥3 in a trap in a mean-field setting with a vanishing factor aN in front of the kinetic energy. The choice aN=N−2∕d is the semi-classical setting and was analysed in great detail in a special, interacting case in [DS21]. Using a version of the well-known Feynman–Kac representation and a further representation in terms of a Poisson point process, we derive precise asymptotics for the reduced one-particle density matrix, implying off-diagonal long-range order (ODLRO, a well-known criterion for Bose–Einstein condensation) for aN above a certain threshold and non-occurrence of ODLRO for aN below that threshold. In particular, we relate the condensate and its total mass to the amount of particles in long loops in the Feynman–Kac formula, the order parameter that Feynman suggested in [Fe53]. For aN≪N−2∕d, we prove that all loops have the minimal length one, and for aN≫N−2∕d we prove 100 percent condensation and identify the distribution of the long-loop lengths as the Poisson–Dirichlet distribution.
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This paper contains a detailed analysis of the Bose condensation of an ideal Bose gas, finite in one dimension and infinite in the other two. Limiting processes, which were treated questionably in previous work, are here re-examined, and some seemingly paradoxical conclusions of the past work are explained. Two new concepts emerge—those of ``further generalized Bose condensation'' (FGBC) and ``anisotropic off-diagonal long-range order'' (AODLRO). The two concepts are shown to be connected to each other, and both are shown to exist at T = 0 for the model discussed.