Ideal Bose-Einstein condensation in the canonical ensemble: exact asymptotic estimates from large deviations

Abstract

In this work we present a large-deviations approach to the calculation of the canonical partition function for free bosons. Three-dimensional Bose-Einstein condensation is studied in the fixed-density ensemble as a function of the dimensionless density $\varrho = \rho \lambda_T^3$, with $\rho=N/L^3$ the standard particle density, $\lambda_T$ the thermal wavelength, $L$ the linear size of the box and $N$ the total number of particles. A large-deviations approach in terms of the dimensionless parameter $\ell=L/\lambda_T$ allows us to provide exact asymptotic estimates of the canonical partition function both above and below the critical density $\varrho_c$ for Bose-Einstein condensation. We show how this approach allows to explicitly account for finite-size effects and how it fully captures the first-order aspects of the transition, allowing us to explicitate its driving mechanism in terms of the competing probabilities of normal and condensed phases. The proposed large-deviations approach allows then to obtain in all regimes explicit and simple analytical expressions, at the leading order in the large parameter $\ell$, for both the average fraction of particles in the ground state, the condensate fraction $\langle n_0(\varrho) \rangle = \langle N_0(\varrho) \rangle/N$, and for its fluctuations, $\sigma_0(\varrho) = \sqrt{\langle N_0^2(\varrho)\rangle - \langle N_0(\varrho)\rangle^2}/N$, retrieving for instance the anomalous scaling $\sigma_0(\varrho)\sim 1/V^{1/3}$ in the condensed regime, $\varrho > \varrho_c$. Our large-deviations asymptotic estimate, by analytically clarifying the mixed-order nature of Bose-Einstein condensation, allows then to reveal the similarity between this transition and other mixed-order transitions, as for instance the localization transition in the Discrete Non-Linear Schr\"odinger Equation.