离散分布与小系统的统计力学
Discrete distributions and statistical mechanics of small systems
AI总结：
研究离散概率分布与小系统统计力学联系，用概率生成函数发展相关理论，应用于巨正则系综描述，提出液体 - 蒸汽相变动力学模型，揭示粒子数分布特性及在Tsallis热力学中的情况，还出现变形离散分布。
AI中文摘要：
我们研究离散概率分布与小系统统计力学之间的联系。利用概率生成函数，我们发展了非负整数值随机变量的幂级数、无限可分和可缩放分布的理论，并引入了在生死过程的平稳解和可缩放无限可分分布中自然出现的马尔可夫分布类。这些结果应用于统计力学中的巨正则系综描述：无限可分性导致相互作用气体的准粒子图像，维里展开与分布的组合相关。提出了液体 - 蒸汽相变的动力学模型，其中临界点处的粒子数分布收敛到离散稳定分布——重整化半群变换的不动点。我们还表明，尽管无限可分性不成立，但粒子数分布的可缩放性在Tsallis非广延热力学中得以保留。此外，变形离散分布，包括作为泊松q变形的负二项分布，在这种情况下自然出现。
英文摘要：
We study connections between discrete probability distributions and the statistical mechanics of small systems. Using probability generating functions, we develop the theory of power series, infinitely divisible, and scalable distributions of non-negative integer-valued random variables, and introduce the class of Markovian distributions that arise naturally in stationary solutions of birth-death processes and in scalable infinitely divisible distributions. These results are applied to the grand canonical ensemble description in statistical mechanics: the infinite divisibility leads to a quasiparticle picture of an interacting gas, and the virial expansion is linked to the combinants of the distribution. A kinetic model of the liquid-vapor phase transition is presented, in which the particle-number distribution at the critical point converges to the Discrete Stable distribution - the fixed point of a renormalization semi-group transformation. We also show that the scalability of the particle-number distribution is preserved in Tsallis non-extensive thermodynamics, even though infinite divisibility fails. Moreover, deformed discrete distributions, including the negative binomial as a q-deformation of the Poisson, arise naturally in this setting.
