‎"KOOPMAN, B. O. - GEORGE D. BIRKHOFF. - THE ERGODIC THEOREM DISCOVERED AND PROVED‎
‎Hamiltonian Systems and Transformation in Hilbert Space. (Koopman) (+) Proof of a Recurrence Theorem for strongly transitive Systems. (Birkhoff) (+) Proof of the Ergodic Theorem. (Birkhoff).‎

‎Easton, PA., Mack printing Compagny, 1931. Royal8vo. Contemp. full cloth. Spine gilt and with gilt lettering. In: ""Proceedings of the National Academy of Sciences of the United States of America"", Vol. 17. VII,710 pp. (Entire volume offered). The papers: pp. 315-318, 650-655 and 656-660.‎

Reference : 48257

‎First editions of these importent papers in statistical mechanics. The so-called Koopman-von Neumann mechanics is a description of classical mechanics in terms of Hilbert space, introduced by Bernard Koopman (the paper offered) and John von Neumann in 1931 and 1932. Ergodicity was introduced by Boltzmann, but the modern theory started from the paper by Koopman, and has been a cornerstone of statistical mechanics since. The ergodic method has found impressive applications in the fields of statistical mechanics, number theory, probability theory, harmonic analysis, and combinatorics.As Koopman and von Neumann demonstrated, a Hilbert space of complex, square integrable wavefunctions can be defined in which classical mechanics can be formulated as an operatorial theory similar to quantum mechanics.Birkhoff's proof (in the third paper offered) of ""the ergodic theorem was deemed as importent as his proof of Poincare's geometric theorem"" (Landmarks Writing in Western Mathematics 1640-1940, p. 877).‎