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

Abstract

Paper is published in Phys. Rev. A 81, 033615 (2010), DOI: 10.1103/PhysRevA.81.033615 We find the universal structure and scaling of the BEC statistics and thermodynamics (Gibbs free energy, average energy, heat capacity) for a mesoscopic canonical-ensemble ideal gas in a trap with arbitrary number of atoms, any volume, and any temperature, including the whole critical region. We identify a universal constraint-cut-off 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 constraintcut-off 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-level and three-level trap models of BEC and find their exact solutions in terms of the cut-off negative binomial distribution (that tends to the cut-off gamma distribution in the continuous limit) and the confluent hypergeometric distribution, respectively. Also, we present an exactly solvable cut-off Gaussian model of BEC in a degenerate interacting gas. All these exact solutions confirm the universality and constraint-cut-off 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 the 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.

Microscopic theory of a phase transition in a critical region: Bose–Einstein condensation in an interacting gas

“…Namely, for a vanishing interaction, when H ′ N ≡ 0 andβ k ≡ 0 in Eqs. (11) and (18), it coincides with a solution for the critical fluctuations in the BEC of a mesoscopic ideal gas [20,21], which is its zeroth-order approximation.…”

“…(14)- (15) amounts to the averages like f (ñ τ1 ,ñ τ2 ) , that is reduced to calculation of a joint distribution ρ n1,n2 of the noncommuting operatorsñ τ1 andñ τ2 . The latter can be done similar to [20,21] via its characteristic function T τ exp(iu 1ñτ1 + iu 2ñτ2 ) , which is equal to…”

“…The calculations are straightforward, but a full analysis is lengthy and will be presented elsewhere. It is based on a distribution of the noncondensate occupation for an ideal gas [20,21],…”

“…They are the step-function θ(N −n) cutoff [20,21] of the Girardeau-Arnowitt operators [37], defined via a transition operatorα k =â † 0 (1 + N −n) −1/2â k ;n = k =0n k . Let us introduce also a Fock space H, which spans only the excited states |n k =0 , and the creation and annihilation operators in H,β † k andβ k , which obey a canonical…”

“…It becomes possible due to the newly developed methods of (a) the nonpolynomial averages and contraction superoperators [15,16], (b) the partial difference (recurrence) equations [17][18][19] (a discrete analog of the partial differential equations) for superoperators, and (c) a characteristic function and cumulant analysis for a joint distribution of the noncommutative observables. They allow us to take into account (I) the constraints in a many-body Hilbert space, which are the integrals of motion prescribed by a broken symmetry in virtue of a Noether's theorem, and constraintcutoff mechanism, responsible for the very existence of a phase transition and its nonanalytical features, [4,20,21] (II) an insufficiency of a grand-canonical-ensemble approximation, which is incorrect in the critical region [2,8] because of averaging over the systems with different numbers of particles, both below and above the critical point, i.e., over the condensed and noncondensed systems at the same time, that implies an error on the order of 100% for any critical function, (III) a necessity to solve the problem for a finite system with a mesoscopic (i.e., large, but finite) number of particles N in order to calculate correctly an anomalously large contribution of the lowest energy levels to the critical fluctuations and to avoid the infrared divergences of the standard thermodynamic-limit approach [5][6][7][8][9][10][11] as well as to resolve a fine structure of the λ-point, (IV) a fact that in the critical region the Dyson-type closed equations do not exist for true Green's functions, but do exist for the partial 1-and 2-contraction superoperators, which reproduce themselves under a contraction.The problem of the critical region and mesoscopic effects is directly related to numerous modern experiments and numerical studies on the BEC of a trapped gas (including BEC on a chip), where N ∼ 10 2 − 10 7 , (see, for example, [22][23][24][25][26][27][28][29][30][31][32][33]) and superfluidit...…”