【2h】

Ergodic theorem ergodic theory and statistical mechanics

机译:遍历定理遍历理论和统计力学

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摘要

This perspective highlights the mean ergodic theorem established by John von Neumann and the pointwise ergodic theorem established by George Birkhoff, proofs of which were published nearly simultaneously in PNAS in 1931 and 1932. These theorems were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60-y-old fundamental problem of the subject—namely, the rationale for the hypothesis that time averages can be set equal to phase averages. The evolution of this problem is traced from the origins of statistical mechanics and Boltzman's ergodic hypothesis to the Ehrenfests' quasi-ergodic hypothesis, and then to the ergodic theorems. We discuss communications between von Neumann and Birkhoff in the Fall of 1931 leading up to the publication of these papers and related issues of priority. These ergodic theorems initiated a new field of mathematical-research called ergodic theory that has thrived ever since, and we discuss some of recent developments in ergodic theory that are relevant for statistical mechanics.
机译:这种观点强调了约翰·冯·诺伊曼建立的平均遍历定理和乔治·伯克霍夫建立的逐点遍历定理,它们的证明几乎同时发表于1931年和1932年的PNAS中。这些定理在数学和统计力学中都具有重要意义。在统计力学中,他们提供了对这个有60年历史的基本问题的关键见解-即,时间平均可以设置为等于相位平均这一假设的理由。这个问题的演变可追溯到统计力学和玻尔兹曼的遍历假说的起源,再到埃伦菲斯特的拟遍历假说,再到遍历的定理。我们讨论了冯·诺伊曼(Ven Neumann)和伯克霍夫(Birkhoff)在1931年秋天之间的交流,这些交流导致这些论文的发表以及相关的优先事项。这些遍历定理开启了自称为遍历理论的数学研究新领域,此后一直蓬勃发展,并且我们讨论了遍历论中与统计力学相关的一些最新发展。

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