## Hamiltonian Systems and Transformations in Hilbert Space (Koopman) AND Proof of the Ergodic Theorem (Birkhoff) AND Proof of the Ergodic Theorem (Birkhoff) in The Proceedings of the National Academy of Sciences Vol. 17, 1931, Washington DC: National Academy of Sciences, pp. 315-318, pp. 650-660, pp. 656-660

Eaton: Mac Printing Co., 1931. 1st Edition. BOUND FIRST EDITION OF BOTH SEMINAL PAPERS IN STATISTICAL MECHANICS – KOOPMAN’S INDIVIDUAL ERGODIC THEOREM, THE BEGINNING OF MODERN ERGODIC THEORY AND BIRKHOFF’S TWO PAPERS, ONE STATING HIS MEAN ERGODIC THEOREM AND THE SECOND, PROVING IT.

Ergodic theory is a branch of mathematics examining dynamical systems with an invariant measure and related problems, its development was motivated by problems of statistical physics. The theorems Koopman and Birkhoff put forth in this volume “were of great significance both in mathematics and in statistical mechanics. In statistical mechanics they provided a key insight into a 60 year old fundamental problem of the subject – namely, the rationale for the hypothesis that time averages can be set equal to phase averages” (Moore, Ergodic Theorem, Ergodic Theory, and Statistical Mechanics, January 2015).

Ergodic theory developed as something of an offshoot from the work on the kinetic theory of gases of Boltzman and Maxwell. “The strategy underlying ergodic theory is to focus on simple yet relevant models to obtain deeper insights about notions that are pertinent to the foundations of statistical mechanics while avoiding unnecessary technical complications. Ergodic theory abstracts away from dynamical associations including forces, potential and kinetic energies, and the like… It focuses on certain structural and dynamical features that are deemed essential to understanding the nature of the seemingly random behaviour of deterministically evolving physical systems. The selected features were carefully integrated, and the end result is a mathematical construct that has proven to be very effective in revealing deep insights that would otherwise have gone unnoticed” (Stanford Encyclopedia of Philosophy).

The modern form of ergodic theory began with the Koopman paper offered here, a work that has been the backbone of statistical mechanics since its appearance. von Neumann’s development of his Hilbert space formulation of quantum mechanics in the late 1920s “inspired Koopman [put forth in this 1931 paper] to develop a Hilbert space formulation of classical statistical mechanics. In both cases, the formula for the time evolution of the state of a system corresponds to a unitary operator that is defined on a Hilbert space; a unitary operator is a type of measure-preserving transformation” (Stanford Encyclopedia of Philosophy). “Koopman’s observation was simultaneously a challenge and a hint. If there is an intimate connection between measure preserving transformations and unitary operators, then the known analytic theory of such operators must surely give some information about the geometric behavior of the transformations” (Halmos, 91).

By October of the same year, von Neumann had the answer; the answer was the mean ergodic theorem; he used Koopman’s innovation to prove what has come to be called the mean ergodic theorem– but he didn’t publish it until 1932. Birkhoff, however, built immediately upon Koopman’s work and published both his ergodic theorem and his proof of his theorem in 1931 and in this volume.

Though von Neumann believed Birkhoff used some of the ideas he had yet to publish and was displeased to be scooped, he also stated that “Birkhoff informed us that he had another proof which showed even somewhat more than mine: Instead of mean convergence, he could prove convergence everywhere excepted on a set of measure 0” (Nadis, A History in Sum). Item #754

CONDITION & DETAILS: Eaton: Mack Printing Company. 4to (10 x 7 inches; 250 x 175mm). Volume has been handsomely rebound in aged brown cloth (identical to original binding). Gilt-lettered and dated at the spine. Solidly and tightly bound. New endpapers. Bright and clean throughout. Near fine condition in every way.

**
Price:
$525.00
**