采煤机点过程与一维硬球模型
Shearer's point process and the hard-sphere model in one dimension
摘要：
我们重新审视一维硬球模型的最小非物理奇点，也称为Tonks气体。我们给出了一个明确的表达式的自由能和减少相关负实逸度和阐述的奇异性的性质：自由能是右连续的，但其衍生物发散。我们得到这些结果在几个新的方式：第一，通过按比例放大的离散解决方案。第二，通过对多布鲁申的配分函数的归纳论证。第三，通过完美的集群扩展，完美地计算了Mayer扩展中的Penrose树。第四，通过显式构造Shearer点过程，得到唯一的具有R-硬核的R-依赖点过程。最后的连接产生明确的和最佳的下限的避免功能的R-依赖的点过程的实数线。
英文摘要：
We revisit the smallest non-physical singularity of the hard-sphere model in one dimension, also known as Tonks gas. We give an explicit expression of the free energy and reduced correlations at negative real fugacity and elaborate the nature of the singularity: the free energy is right-continuous, but its derivative diverges. We derive these results in several novel ways: First, by scaling up the discrete solution. Second, by an inductive argument on the partition function à la Dobrushin. Third, by a perfect cluster expansion counting the Penrose trees in the Mayer expansion perfectly. Fourth, by an explicit construction of Shearer's point process, the unique R-dependent point process with an R-hard-core. The last connection yields explicit and optimal lower bounds on the avoidance function of R-dependent point processes on the real line.
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