正则系综中的理想玻色-爱因斯坦凝聚：基于大偏差的精确渐近估计

中文摘要 AI 辅助

本工作提出一种大偏差方法，用于计算自由玻色子的正则配分函数。在固定密度系综中研究三维玻色-爱因斯坦凝聚，无量纲密度为 $\u03f1 = \rho \lambda_T^3$，其中 $\u03c1=N/L^3$ 为标准粒子密度，$\u03bb_T$ 为热波长，$L$ 为盒子的线性尺寸，$N$ 为粒子总数。基于无量纲参数 $\u2113=L/\lambda_T$ 的大偏差方法，可在玻色-爱因斯坦凝聚临界密度 $\u03f1_c$ 上下，给出正则配分函数的精确渐近估计。研究表明该方法可明确考虑有限尺寸效应，完整捕捉相变的一阶特性，还能通过正常相与凝聚相的竞争概率阐明相变驱动机制。所提大偏差方法在所有区域均可得到简洁的解析表达式，在大参数 $\u2113$ 的主导阶下，分别给出基态粒子的平均分数（即凝聚分数 $\u27e8 n_0(\u03f1) \u27e9 = \u27e8 N_0(\u03f1) \u27e9 / N$）及其涨落（$\u03c3_0(\u03f1) = \sqrt{\u27e8 N_0^2(\u03f1) \u27e9 - \u27e8 N_0(\u03f1) \u27e9^2} / N$），例如在凝聚区（$\u03f1 > \u03f1_c$）得到异常标度 $\u03c3_0(\u03f1) \sim 1/V^{1/3}$。该大偏差渐近估计通过解析阐明玻色-爱因斯坦凝聚的混合阶特性，揭示其与其他混合阶相变（如离散非线性薛定谔方程中的局域化相变）的相似性。

英文摘要

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 $\varrho = ρλ_T^3$, with $ρ=N/L^3$ the standard particle density, $λ_T$ 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 $\ell=L/λ_T$ 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, $σ_0(\varrho) = \sqrt{\langle N_0^2(\varrho)\rangle - \langle N_0(\varrho)\rangle^2}/N$, retrieving for instance the anomalous scaling $σ_0(\varrho)\sim 1/V^{1/3}$ 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 Schrödinger Equation.

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19 pages, 4 figures