论文标题
通过渐近膨胀和光谱$ζ$ functions的理想玻色气体热力学极限
The thermodynamic limit of an ideal Bose gas by asymptotic expansions and spectral $ζ$-functions
论文作者
论文摘要
我们分析了理想的，非权利主义的玻色气体的热力学极限 - 以各向同性谐波电位的开放陷阱极限为模型，并特别强调了Bose-Einstein凝结现象。这是通过使用大电位的渐近扩展来实现的，该扩展是由$ζ$ regulination技术得出的。因此，我们可以证明，该扩展的奇异性结构直接与系统的相结构交织在一起：在非构敏阶段，扩展具有类似于常规的热核扩展的形式。这样，热力学可观察物是可以直接计算的。相比之下，膨胀表现出高于临界密度以上的无限顺序的奇异性，并且需要对化学电位的重新归一化，以确保定义明确的热力学可观察物。此外，重新归一化的程序迫使系统表现出凝结。此外，我们表明，热力学极限的特征（如临界密度或内部能量）完全编码在渐近扩张的系数中。
We analyze the thermodynamic limit - modeled as the open-trap limit of an isotropic harmonic potential - of an ideal, non-relativistic Bose gas with a special emphasis on the phenomenon of Bose-Einstein condensation. This is accomplished by the use of an asymptotic expansion of the grand potential, which is derived by $ζ$-regularization techniques. Herewith we can show, that the singularity structure of this expansion is directly interwoven with the phase structure of the system: In the non-condensation phase the expansion has a form that resembles usual heat kernel expansions. By this, thermodynamic observables are directly calculable. In contrast, the expansion exhibits a singularity of infinite order above a critical density and a renormalization of the chemical potential is needed to ensure well-defined thermodynamic observables. Furthermore, the renormalization procedure forces the system to exhibit condensation. In addition, we show that characteristic features of the thermodynamic limit, like the critical density or the internal energy, are entirely encoded in the coefficients of the asymptotic expansion.
