Analytical theory of mesoscopic Bose-Einstein condensation in an ideal gas

Journal Article · Mon Mar 15 00:00:00 EDT 2010 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.81.033615· OSTI ID:21413306

Kocharovsky, Vitaly V ^[1]; Kocharovsky, Vladimir V ^[2]

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1. Department of Physics, Texas A and M University, College Station, Texas 77843-4242 (United States)
2. Institute of Applied Physics, Russian Academy of Science, RU-603950 Nizhny Novgorod (Russian Federation)

We find the universal structure and scaling of the Bose-Einstein condensation (BEC) statistics and thermodynamics (Gibbs free energy, average energy, heat capacity) for a mesoscopic canonical-ensemble ideal gas in a trap with an arbitrary number of atoms, any volume, and any temperature, including the whole critical region. We identify a universal constraint-cutoff mechanism that makes BEC fluctuations strongly non-Gaussian and is responsible for all unusual critical phenomena of the BEC phase transition in the ideal gas. The main result is an analytical solution to the problem of critical phenomena. It is derived by, first, calculating analytically the universal probability distribution of the noncondensate occupation, or a Landau function, and then using it for the analytical calculation of the universal functions for the particular physical quantities via the exact formulas which express the constraint-cutoff mechanism. We find asymptotics of that analytical solution as well as its simple analytical approximations which describe the universal structure of the critical region in terms of the parabolic cylinder or confluent hypergeometric functions. The obtained results for the order parameter, all higher-order moments of BEC fluctuations, and thermodynamic quantities perfectly match the known asymptotics outside the critical region for both low and high temperature limits. We suggest two- and three-level trap models of BEC and find their exact solutions in terms of the cutoff negative binomial distribution (which tends to the cutoff gamma distribution in the continuous limit) and the confluent hypergeometric distribution, respectively. Also, we present an exactly solvable cutoff Gaussian model of BEC in a degenerate interacting gas. All these exact solutions confirm the universality and constraint-cutoff origin of the strongly non-Gaussian BEC statistics. We introduce a regular refinement scheme for the condensate statistics approximations on the basis of the infrared universality of higher-order cumulants and the method of superposition and show how to model BEC statistics in the actual traps. In particular, we find that the three-level trap model with matching the first four or five cumulants is enough to yield remarkably accurate results for all interesting quantities in the whole critical region. We derive an exact multinomial expansion for the noncondensate occupation probability distribution and find its high-temperature asymptotics (Poisson distribution) and corrections to it. Finally, we demonstrate that the critical exponents and a few known terms of the Taylor expansion of the universal functions, which were calculated previously from fitting the finite-size simulations within the phenomenological renormalization-group theory, can be easily obtained from the presented full analytical solutions for the mesoscopic BEC as certain approximations in the close vicinity of the critical point.

• OSTI ID: 21413306
• Journal Information: Physical Review. A, Vol. 81, Issue 3; Other Information: DOI: 10.1103/PhysRevA.81.033615; (c) 2010 The American Physical Society; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANALYTICAL SOLUTION
APPROXIMATIONS
ASYMPTOTIC SOLUTIONS
ATOMS
BOSE-EINSTEIN CONDENSATION
DISTRIBUTION
EXACT SOLUTIONS
EXPANSION
FLUCTUATIONS
FREE ENTHALPY
GROUP THEORY
HYPERGEOMETRIC FUNCTIONS
ORDER PARAMETERS
PHASE TRANSFORMATIONS
SPECIFIC HEAT
STATISTICS
TRAPS
CALCULATION METHODS
DIMENSIONLESS NUMBERS
ENERGY
FUNCTIONS
MATHEMATICAL SOLUTIONS
MATHEMATICS
PHYSICAL PROPERTIES
THERMODYNAMIC PROPERTIES
VARIATIONS