scholarly journals Ergodic theorem, ergodic theory, and statistical mechanics

2015 ◽  
Vol 112 (7) ◽  
pp. 1907-1911 ◽  
Author(s):  
Calvin C. Moore
Keyword(s):  
Statistical Mechanics ◽  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Fundamental Problem ◽  
Ergodic Theorems ◽  
Von Neumann ◽  
Ergodic Hypothesis ◽  
Recent Developments ◽  
The Subject ◽  
Time Averages

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.

scholarly journals Actions of lattices in Sp (1, n)

1989 ◽  
Vol 9 (2) ◽  
pp. 221-237 ◽  
Author(s):  
Michael Cowling ◽  
Robert J. Zimmer
Keyword(s):  
Ergodic Theory ◽  
Von Neumann Algebras ◽  
Von Neumann ◽  
And Topology

AbstractWe study questions concerning the ergodic theory, von Neumann algebras, geometry, and topology of actions of lattices in Sp (1, n).

Von Neumann Algebras and Ergodic Theory of Group Actions

2008 ◽  
pp. 2763-2814
Author(s):  
Dietmar Bisch ◽  
Damien Gaboriau ◽  
Vaughan Jones ◽  
Sorin Popa
Keyword(s):  
Ergodic Theory ◽  
Von Neumann Algebras ◽  
Group Actions ◽  
Von Neumann

scholarly journals Topological dynamics and ergodic theory

1987 ◽  
Vol 7 (1) ◽  
pp. 25-47 ◽  
Author(s):  
Robert Ellis
Keyword(s):  
Discrete Spectrum ◽  
Ergodic Theory ◽  
General Setting ◽  
Topological Dynamics ◽  
Von Neumann ◽  
Ergodic Processes ◽  
Neumann Theorem ◽  
Von Neumann Theorem

AbstractIt is shown that when viewed properly some concepts in topological dynamics and ergodic theory are not merely analogous but equivalent. Also the Mackey-Halmos-von Neumann theorem on ergodic processes with discrete spectrum is generalized and an account of the Mackey-Zimmer theory of minimal cocycles is given in a more general setting.

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