arXiv:cond-mat/0001422 · Journal of Physics B Atomic Molecular and Optical Physics

Transition temperature of the weakly interacting Bose gas: perturbative solution of the crossover equations in the canonical ensemble

Martin Wilkens, Fabrizio Illuminati, Meret Krämer

Published Oct 5, 2000• 30 citations

Abstract

We compute the critical temperature T_c of a weakly interacting uniform Bose gas in the canonical ensemble, extending the criterion of condensation provided by the counting statistics for the uniform ideal gas. Using ordinary perturbation theory, we find in first order $(T_c-T_c^0)/T_c^0 = -0.93 aρ^{1/3}$, where T_c^0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and $ρ$ is the particle number density.

DOI: 10.1088/0953-4075/33/20/10j

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arXiv:cond-mat/0001422 · Journal of Physics B Atomic Molecular and Optical Physics

Transition temperature of the weakly interacting Bose gas: perturbative solution of the crossover equations in the canonical ensemble

Martin Wilkens, Fabrizio Illuminati, Meret Krämer

Published Oct 5, 2000• 30 citations

arXiv HTML PDF

Abstract

(T_c-T_c^0)/T_c^0 = -0.93 aρ^{1/3}

, where T_c^0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and

ρ

is the particle number density.

DOI: 10.1088/0953-4075/33/20/10j

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(T_c-T_c^0)/T_c^0 = -0.93 aρ^{1/3}

, where T_c^0 is the transition temperature of the corresponding ideal Bose gas, a is the scattering length, and

ρ

is the particle number density.

The shift of the Bose-Einstein condensation temperature for a homogenous weakly interacting Bose gas in leading order in the scattering length

a

is computed for given particle density

n

. Variational perturbation theory is used to resum the corresponding perturbative series for $\ensuremath{\Delta}⟨{\ensuremath{\phi}}^{2}⟩∕Nu$ in a classical three-dimensional scalar field theory with coupling

u

and where the physical case of

N=2

field components is generalized to arbitrary

N

. Our results for

N=1,2,4

are in agreement with recent Monte Carlo simulations; for

N=2

, we obtain $\ensuremath{\Delta}{T}_{c}∕{T}_{0}=(1.27\ifmmode\pm\else\textpm\fi{}0.11)a{n}^{1∕3}$. We use seven-loop perturbative coefficients, extending earlier work by one loop order.

We study condensate statistics and thermodynamics of weakly interacting Bose gas with a fixed total number

N

of particles in a cubic box. We find the exact recursion relation for the canonical ensemble partition function. Using this relation, we calculate the distribution function of condensate particles for

N=200

. We also calculate the distribution function based on multinomial expansion of the characteristic function. Similar to the ideal gas, both approaches give exact statistical moments for all temperatures in the framework of Bogoliubov model. We compare them with the results of unconstraint canonical ensemble quasiparticle formalism and the hybrid master equation approach. The present recursion relation can be used for any external potential and boundary conditions. We investigate the temperature dependence of the first few statistical moments of condensate fluctuations as well as thermodynamic potentials and heat capacity analytically and numerically in the whole temperature range.