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Journal of Modern Optics

Beyond Gibbs' method in statistical physics: theorem on non-polynomial averages and non-perturbation in fluctuations theory of Bose–Einstein condensation

Vl. V. Kocharovsky, Vl. V. Kocharovsky

Published Nov 10, 2007• 9 citations
Publisher

Abstract

General theorem on non-polynomial averages in statistical physics, including anomalous averages, is found. It generalizes a well-known Wick's theorem and, in particular, yields a fully quantum kinetic theory of Bose–Einstein condensation. A non-perturbation-in-fluctuations BEC rate is presented.
DOI
10.1080/09500340701639607

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Abstract In Ref. Gaudin (Nuclear Physics 15 , 89 1960), Gaudin established a generalized version of Wick’s theorem at finite temperature for systems described by a diagonal quadratic Hamiltonian. In this work, we extend Gaudin’s proof to statistical density operators generated by a general quadratic Hamiltonian, prior to diagonalization, as commonly encountered in interacting Bose systems. We show explicitly that the resulting thermal contractions depend on the Bogoliubov transformation parameters and on the quasiparticle spectrum, providing a transparent framework for evaluating thermal averages of arbitrary operator products. As an illustration of the usefulness of this generalized formulation, we evaluate particle-number fluctuations in a homogeneous, weakly interacting Bose gas below the Bose–Einstein condensation temperature. We show that particle-number fluctuations are generically non-Poissonian as a consequence of interaction-induced correlations and quantum depletion, with the Poissonian limit recovered only for an ideal Bose gas at zero temperature. Our results clarify the role of quadratic Hamiltonians and Bogoliubov quasiparticles in shaping fluctuation statistics and highlight the generalized Wick’s theorem as a controlled tool for computing correlation functions in interacting quantum gases.
Bose-Einstein condensation (BEC) represents a remarkable phase transition, characterized by the formation of a single quantum subsystem. As a result, the statistical properties of the condensate are highly unique. In the case of a Bose gas, while the mean number of condensed atoms is independent of the choice of statistical ensemble, the microcanonical, canonical (CN), or grand CN (GC) variances differ significantly among these ensembles. In this paper, we review the progress made over the past 30 years in studying the statistical fluctuations of BECs. Focusing primarily on the ideal Bose gas, we emphasize the inequivalence of the Gibbs statistical ensembles and examine various approaches to this problem. These approaches include explicit analytic results for primarily one-dimensional systems, methods based on recurrence relations, asymptotic results for large numbers of particles, techniques derived from laser theory, and methods involving the construction of statistical ensembles via stochastic processes, such as the Metropolis algorithm. We also discuss the less thoroughly resolved problem of the statistical behavior of weakly interacting Bose gases. In particular, we elaborate on our stochastic approach, known as the hybrid sampling method. The experimental aspect of this field has gained renewed interest, especially following groundbreaking recent measurements of condensate fluctuations. These advancements were enabled by unprecedented control over the total number of atoms in each experimental realization. Additionally, we discuss the fluctuations in photonic condensates as an illustrative example of GC fluctuations. Finally, we briefly consider the future directions for research in the field of condensate statistics.
Wick's theorem is combined with a cluster composition to derive a generalized statistical Wick's theorem for a non-equilibrium system with initial correlations. This enables a diagrammatic time-dependent perturbation theory which is a simple extension of the usual one to be developed for time-path causal statistical Green functions and their matrix equivalents. This provides a complete non-equilibrium theory. Reduction of diagrams and equations of motion for the Green functions are also discussed.
We calculate the Bose-Einstein-condensate (BEC) occupation statistics vs the interparticle interaction in a dilute gas with a nonuniform condensate in a box trap within the Bogoliubov approach. The results are compared against the previously found BEC-occupation statistics in (i) an ideal gas and (ii) a weakly interacting gas with a uniform condensate. In particular, we reveal and explicitly describe an appearance of a nontrivial transition from the ideal gas to the Thomas-Fermi regime. The results include finding the main regimes of the BEC statistics---the anomalous non-Gaussian thermally dominated fluctuations and the Gaussian quantum-dominated fluctuations---as well as a crossover between them and their manifestations in a mesoscopic system. Remarkably, we show that the effect of the boundary conditions, imposed at the box trap, on the BEC fluctuations does not vanish in the thermodynamic limit of a macroscopic system even in the presence of the interparticle interactions. Finally, we discuss a challenging problem of an experimental verification of the theory of the BEC fluctuations addressing a much deeper level of the many-body statistical physics than usually studied quantities related to the mean condensate occupation.