正则系综中占据数的相关性及其对一维谐波阱中BEC的应用
Correlations of occupation numbers in the canonical ensemble and application to BEC in a 1D harmonic trap
摘要：
研究了正则系综中$N$非相互作用全同玻色子或费米子的统计性质。我们推导出几个一般表示的$p$点相关函数的占领数$\overline{n_1\cdots n_p}$。我们证明，它可以表示为两个$p\times p$决定因素的比率，涉及（规范）平均职业$\overline{n_1}$，.，$\overline{n_p}$，它本身可以方便地用$k$ -体配分函数（用$k\leq N$）表示。我们与对称函数的理论联系起来，得到了相关函数用舒尔函数表示的表达式。我们的研究结果通过重新讨论一维谐振势阱中的玻色-爱因斯坦凝聚问题得到了解析结果。得到了占据数的矩以及基态和激发态占据数的相关性。在由量子关联控制的温度范围内，基态占有率的分布符合截断的Gumbel定律。当温度远小于玻色-爱因斯坦温度时，得到了描述极值统计的Gumbel定律。
英文摘要：
We study statistical properties of $N$ non-interacting identical bosons or fermions in the canonical ensemble. We derive several general representations for the $p$-point correlation function of occupation numbers $\overline{n_1\cdots n_p}$. We demonstrate that it can be expressed as a ratio of two $p\times p$ determinants involving the (canonical) mean occupations $\overline{n_1}$, ..., $\overline{n_p}$, which can themselves be conveniently expressed in terms of the $k$-body partition functions (with $k\leq N$). We draw some connection with the theory of symmetric functions, and obtain an expression of the correlation function in terms of Schur functions. Our findings are illustrated by revisiting the problem of Bose-Einstein condensation in a 1D harmonic trap, for which we get analytical results. We get the moments of the occupation numbers and the correlation between ground state and excited state occupancies. In the temperature regime dominated by quantum correlations, the distribution of the ground state occupancy is shown to be a truncated Gumbel law. The Gumbel law, describing extreme value statistics, is obtained when the temperature is much smaller than the Bose-Einstein temperature.
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