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Journal of Physics B Atomic Molecular and Optical Physics

Phase transitions in ideal and weakly interacting Bose gases with a finite number of particles confined in a box

Jian-hui Wang, Yong-li Ma

Published Dec 4, 2009• 9 citations
Publisher

Abstract

We generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane and approach the classifications of phase transitions in ideal and weakly interacting Bose gases of a finite number of particles, confined in a cubic box of volume L3 with different boundary conditions. For this finite ideal Bose system, by extending the classification parameters to all regions, we predict that the phase transition for periodic boundary conditions is of second order, while the transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the effects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on interaction strengths and particle numbers.

DOI
10.1088/0953-4075/42/24/245301

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We generalize the scheme to characterize phase transitions of finite systems in a complex temperature plane and approach the classifications of phase transitions in ideal and weakly interacting Bose gases of a finite number of particles, confined in a cubic box of volume L3 with different boundary conditions. For this finite ideal Bose system, by extending the classification parameters to all regions, we predict that the phase transition for periodic boundary conditions is of second order, while the transition in Dirichlet boundary conditions is of first order. For a weakly interacting Bose gas with periodic boundary conditions, we discuss the effects of finite particle numbers and inter-particle interactions on the nature of the phase transitions. We show that this homogenous weakly interacting Bose gas undergoes a second-order phase transition, which is in accordance with universality arguments for infinite systems. We also discuss the dependence of transition temperature on interaction strengths and particle numbers.
The thermodynamic properties and finite-size scaling of weakly interacting Bose gases with a finite number of particles confined in a cubic box of volume L3{L}^{3}L3 with periodic boundary conditions are investigated theoretically based on an exact recursion relation for the canonical-ensemble partition function. In a similar manner to the case of an ideal Bose gas, the recursive scheme of the interacting finite system at any temperatures is established in the framework of the Bogoliubov theory. The temperature dependence of the condensate fraction and specific heat with different particles and interactions is obtained numerically. By defining two different transition temperatures of the finite systems, the effect of interactions on the transition temperatures yields the different and nonmonotonous behaviors. The finite-size scaling condensate fraction at the transition temperature for the systems is also discussed. It is shown that the calculated finite-size scaling is universally independent on the various system sizes and transition temperatures.
We conduct a rigorous investigation into the thermodynamic instability of an ideal Bose gas confined in a cubic box, without assuming a thermodynamic limit or a continuous approximation. Based on the exact expression of the canonical partition function, we perform numerical computations up to 106{10}^{6}106 particles. We report that if the number of particles is equal to or greater than a certain critical value, which turns out to be 761676167616, the ideal Bose gas subject to the Dirichlet boundary condition reveals a thermodynamic instability. Accordingly, we demonstrate that a system consisting of a finite number of particles can exhibit a discontinuous phase transition that features a genuine mathematical singularity, provided we keep not volume but pressure constant. The specific number, 7616,7616,7616, can be regarded as a characteristic number of a ``cube,'' which is the geometric shape of the box.
We investigate the probability distribution that governs the number of ground-state particles in a partially condensed ideal Bose gas confined to a cubic volume within the canonical ensemble. Imposing either periodic or Dirichlet boundary conditions, we derive asymptotic expressions for all its cumulants. Whereas the condensation temperature becomes independent of the boundary conditions in the large-system limit, as implied by Weyl's theorem, the fluctuation of the number of condensate particles and all higher cumulants remain sensitive to the boundary conditions even in that limit. The implications of these findings for weakly interacting Bose gases are briefly discussed.
We consider a homogeneous Bose gas of particles with an attractive interaction in an elongated three-dimensional box with periodic boundary conditions. Mean-field theory predicts for this system a spontaneous breaking of the translational symmetry at a certain value of the interaction strength. We show that at this point a second-order quantum phase transition occurs. We investigate the system in the vicinity of the critical point using Bogoliubov theory and a continuous description, that allows us to analyze quantum fluctuations in the system even when the Bogoliubov approach breaks down.
We present a detailed description of a classification scheme for phase transitions in finite systems based on the distribution of Fisher zeros of the canonical partition function in the complex temperature plane. We apply this scheme to finite Bose systems in power-law traps within a semi-analytic approach with a continuous one-particle density of states $\ensuremath{\Omega}(E)\ensuremath{\sim}{E}^{d\ensuremath{-}1}$ for different values of d and to a three-dimensional harmonically confined ideal Bose gas with discrete energy levels. Our results indicate that the order of the Bose-Einstein condensation phase transition sensitively depends on the confining potential.
Within a canonical ensemble treatment, we study the finite-size scaling behaviour of the specific heat near the critical region for the weakly interacting Bose gases, and obtain the specific heat exponent α and correlation length exponent ν which agree well with experimental data and previous theoretical predictions. For the small and fixed interaction parameter an 1/3 with a being the scattering length and n the particle number density, we report for the first time that the dimensionless specific heats per particle c ( t , N ) for various system sizes ( L = ( N / n ) 1/3 in a square box potential) and reduced temperatures ( t = T / T c − 1) collapse onto a single form, and the borders of universality validity are obtained numerically. From scaling arguments, we study the interaction-induced shift of the transition temperature (Δ T c = T c − T 0 c ) and find that Δ T c / T 0 c = ban 1/3 with b = 0.42 ± 0.05, where T 0 c is the ideal-gas thermodynamic critical temperature. We also compare our findings with experimental and theoretical results for in cubic lattices, showing the very different behaviours for the two systems.