标题： 与离散高斯-泊松分布相关的新的特殊函数以及具有居里-韦斯相互作用的细胞模型的一些物理性质 显示英文标题

标题： A new special function related to a discrete Gauss-Poisson distribution and some physics of the cell model with Curie-Weiss interactions

摘要： 受统计物理中先前研究的启发[特别是Kozitsky等人，具有二元相互作用的Curie-Weiss系统中的相变，Condens。Matter Phys。23, 23502 (2020)]，我们引入了一个在$\mathbb N_0$上具有支撑的离散高斯-泊松概率分布函数\begin{equation}\label{GPD}\tag{A1} p_{GP}(n ;z,r)=\left[R(r;z)\right]^{-1}\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2} \end{equation}，参数为$z\in\mathbb R$和$r\in\mathbb R_+$。概率质量函数$p_{GP}(n ;z,r)$由特殊函数$R(r;z)$归一化，该函数由无限和\begin{equation}\label{R}\tag{A2} R(r;z)=\sum_{n=0}^\infty\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2}, \end{equation}给出，具有极其有趣的数学性质。 We present an asymptotic estimate $R^{(\rm as)}(r;z\gg1)$ for the function $R(r;z)$ with large arguments $z$, along with similar formulas for its logarithm and logarithmic derivative. These functions exhibit very interesting oscillatory behavior around their asymptotics, for parameters $r$ above some threshold value $r^*$. Some implications of our findings are discussed in the context of the Curie-Weiss cell model of simple fluids. 显示英文摘要

摘要： Inspired by previous studies in statistical physics [see, in particular, Kozitsky at al., A phase transition in a Curie-Weiss system with binary interactions, Condens. Matter Phys. 23, 23502 (2020)] we introduce a discrete Gauss-Poisson probability distribution function \begin{equation}\label{GPD}\tag{A1} p_{GP}(n ;z,r)=\left[R(r;z)\right]^{-1}\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2} \end{equation} with support on $\mathbb N_0$ and parameters $z\in\mathbb R$ and $r\in\mathbb R_+$. The probability mass function $p_{GP}(n ;z,r)$ is normalized by the special function $R(r;z)$, given by the infinite sum \begin{equation}\label{R}\tag{A2} R(r;z)=\sum_{n=0}^\infty\frac{\mbox{e}^{zn}}{n!}\,\mbox{e}^{-\frac 12\,rn^2}, \end{equation} possessing extremely intersting mathematical properties. We present an asymptotic estimate $R^{(\rm as)}(r;z\gg1)$ for the function $R(r;z)$ with large arguments $z$, along with similar formulas for its logarithm and logarithmic derivative. These functions exhibit very interesting oscillatory behavior around their asymptotics, for parameters $r$ above some threshold value $r^*$. Some implications of our findings are discussed in the context of the Curie-Weiss cell model of simple fluids.