多组分Tonks气体的集团展开和维里展开
Cluster and virial expansions for the multi-species Tonks gas
摘要：
我们考虑不同长度$\ell_k$的非重叠杆在$\mathbb{R}$或$\mathbb{Z}$中运动的混合。我们的主要结果是压力按活度$z_k$和密度$ρ_k$展开的充要收敛准则。这提供了一个明确的例子来测试已知的集群扩展标准，并说明，对于非负相互作用，维里扩展可以收敛在一个域比活动扩展大得多。此外，我们给出明确的公式，推广了众所周知的非重叠杆和标记根树之间的关系。我们还证明，对于某些活动的选择，系统可以经历类似于零程过程的凝聚过渡。关键工具是压力的定点方程。
英文摘要：
We consider a mixture of non-overlapping rods of different lengths $\ell_k$ moving in $\mathbb{R}$ or $\mathbb{Z}$. Our main result are necessary and sufficient convergence criteria for the expansion of the pressure in terms of the activities $z_k$ and the densities $ρ_k$. This provides an explicit example against which to test known cluster expansion criteria, and illustrates that for non-negative interactions, the virial expansion can converge in a domain much larger than the activity expansion. In addition, we give explicit formulas that generalize the well-known relation between non-overlapping rods and labelled rooted trees. We also prove that for certain choices of the activities, the system can undergo a condensation transition akin to that of the zero-range process. The key tool is a fixed point equation for the pressure.
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