Ergodic theorem, ergodic theory, and statistical mechanics
Ergodic theorem, ergodic theory, and statistical mechanics
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DOI：
10.1073/pnas.1421798112
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发表时间：
2015-02-17
影响因子：
11.1
通讯作者：
Moore, Calvin C.
中科院分区：
文献类型：
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作者：
Moore, Calvin C.
这个观点突出了约翰·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在1931年和1932年的《美国国家科学院院刊》上。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，他们提供了对该学科一个60年前的基本问题的关键洞察--即，可以将时间平均设置为等于相位平均的假设的基本原理。这个问题的演变可以追溯到统计力学和玻尔兹曼的遍历假说的起源，到Ehrenfest的准遍历假说，再到遍历定理。我们讨论了冯·诺依曼和伯克霍夫在1931年秋天之间的交流，导致了这些论文的出版以及相关的优先事项。这些遍历定理开创了一个新的数学研究领域，称为遍历理论，从那以后一直蓬勃发展，我们讨论了与统计力学相关的遍历理论的一些最新发展。
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.