States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures
States of classical statistical mechanical systems of infinitely many particles. II. Characterization of correlation measures
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无限多个粒子的经典统计力学系统的状态。
DOI：
10.1007/bf00251602
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发表时间：
1975
影响因子：
2.5
通讯作者：
A. Lenard
中科院分区：
文献类型：
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作者：
A. Lenard
本文致力于解决点粒子经典统计力学中自然产生的关联测度的刻画问题。在无限粒子位形空间X上，一个相关测度ρ必须与一个概率测度μ（不一定是唯一的）相关，公式为μ（H）= NH（）d μ，其中{NH}是某个整值随机变量族。我们证明了有三个条件，即（S）对称性，（P）正性和（N）归一化，这三个条件是ρ成为相关测度的充分必要条件。主要的运算条件是（P），它表示对于任何函数φ，S φ（φ）d ρ <$0必须成立，其中φ → S φ是一个线性算子，我们研究了它的性质。条件（P）给出了一类由某些集合的ρ-测度所满足的不等式。这个理论也被推广到当有一组平移作用在单粒子空间中时的情况，这时我们关心的是相对于这个群不变的测度ρ和μ。
This paper is devoted to the solution of the problem of characterizing correlation measures arising naturally in classical statistical mechanics of point particles. A correlation measure ρ must be related to a (not necessarily unique) probability measure μ over an infinite particle configuration space X by the formula μ(H)=∝NH(ξ)dμ where {NH} is a certain family of integer valued random variables. We prove that there are three conditions, namely (S) symmetry, (P) positivity, and (N) normalization, which together are sufficient as well as necessary for ρ to be a correlation measure. The main operative condition is (P), which says that ξφ(x)dρ≧0 must hold for every function φ for which Sφ(ξ)≧0 identically for ξ ε X, where φ → Sφ is a certain linear operator whose properties we study. Condition (P) gives rise to a large class of inequalities satisfied by the ρ-measures of certain sets. The theory is also generalized to the case when there is a group of translations acting in the one-particle space, the concern then being with measures ρ as well as μ that are invariant with respect to the group.