Let z be the activity of point particles described by classical equilibrium statistical mechanics in $$\mathbf{R}^\nu $$ R ν . The correlation functions $$\rho ^z(x_1,\dots ,x_k)$$ ρ z ( x 1 , ⋯ , x k ) denote the probability densities of finding k particles at $$x_1,\dots ,x_k$$ x 1 , ⋯ , x k . Letting $$\phi ^z(x_1,\dots ,x_k)$$ ϕ z ( x 1 , ⋯ , x k ) be the cluster functions corresponding to the $$\rho ^z(x_1,\dots ,x_k)/z^k$$ ρ z ( x 1 , ⋯ , x k ) / z k we prove identities of the type $$\begin{aligned}&\phi ^{z_0+z'}(x_1,\dots ,x_k)\\&\quad =\sum _{n=0}^\infty {z'^n\over n!}\int dx_{k+1}\dots \int dx_{k+n}\,\phi ^{z_0}(x_1,\dots ,x_{k+n}) \end{aligned}$$ ϕ z 0 + z ′ ( x 1 , ⋯ , x k ) = ∑ n = 0 ∞ z ′ n n ! ∫ d x k + 1 ⋯ ∫ d x k + n ϕ z 0 ( x 1 , ⋯ , x k + n ) It is then non-rigorously argued that, assuming a suitable cluster property (decay of correlations) for a crystal state, the pressure and the translation invariant correlation functions $$\rho ^z(x_1,\dots ,x_k)$$ ρ z ( x 1 , ⋯ , x k ) are real analytic functions of z .

中文翻译：

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经典统计力学中相关函数的恒等式和晶态问题

设 z 为 $$\mathbf{R}^\nu $$ R ν 中经典平衡统计力学描述的点粒子的活动。相关函数 $$\rho ^z(x_1,\dots ,x_k)$$ ρ z ( x 1 , ⋯ , xk ) 表示在 $$x_1,\dots ,x_k$$ x 1 处找到 k 个粒子的概率密度, ⋯ , xk 。令 $$\phi ^z(x_1,\dots ,x_k)$$ ϕ z ( x 1 , ⋯ , xk ) 为对应于 $$\rho ^z(x_1,\dots ,x_k)/z 的聚类函数^k$$ ρ z ( x 1 , ⋯ , xk ) / zk 我们证明 $$\begin{aligned}&\phi ^{z_0+z'}(x_1,\dots ,x_k)\\& 类型的恒等式\quad =\sum _{n=0}^\infty {z'^n\over n!}\int dx_{k+1}\dots \int dx_{k+n}\,\phi ^{z_0} (x_1,\dots ,x_{k+n}) \end{aligned}$$ ϕ z 0 + z ′ ( x 1 , ⋯ , xk ) = ∑ n = 0 ∞ z ′ nn ！∫ dxk + 1 ⋯ ∫ dxk + n ϕ z 0 ( x 1 , ⋯ , xk + n ) 然后不严谨地论证，