Birkhoff G D (16 results)

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Condition: Poor. This is an ex-library book and may have the usual library/used-book markings inside.This book has hardback covers. In poor condition, suitable as a reading copy. No dust jacket. Please note the Image in this listing is a stock photo and may not match the covers of the actual item,750grams, ISBN:

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Condition: New. pp. 236.

Original wrappers. Condition: Very Good. First edition. IMPORTANT PAPERS - IN BOTH OFFPRINT AND JOURNAL FORMATS - RELATED TO VON NEUMANN'S PROOF OF THE ERGODIC THEOREM, A FUNDAMENTAL CONTRIBUTION TO OUR UNDERSTANDING OF THE FOUNDATIONS OF STATISTICAL MECHANICS AND THE ORIGINS OF THE SECOND LAW OF THERMODYNAMICS. In 1877 the great physicist Ludwig Boltzmann published a paper that sought to explain why systems tend toward maximum-entropy equilibrium states as predicted by the Second Law of Thermodynamics. The conceptual essence of the 1877 paper was this: a system - say, a gas in a container - can exist in any of an inconceivably large number of "states," each differing from all others in the precise position or velocity of at least one molecule. In modern terminology, each such distinguishable disposition of the molecules of a physical system is called a "microstate". But enormous numbers of microstates are equivalent in macroscopic terms - although they differ in terms of the precise locations and velocities of specific molecules, they look the same from the perspective of bulk thermodynamic variables such as temperature and pressure. In other words, such microstates are part of the same "macrostate." The key to Boltzmann's 1877 analysis was his recognition that the maximum-entropy state of a system - i.e., its equilibrium state - is precisely the macrostate that is consistent with the largest number of microstates. (This can be shown by a rather elementary application of combinatorial analysis, which Boltzmann undertook in the 1877 paper.) Accordingly, it is overwhelmingly more probable than not that a system - as it transitions from one microstate to the next - will eventually wind up in one of the microstates corresponding to the equilibrium macrostate. There was, however, important gap in Boltzmann's magisterial 1877 analysis - a gap that Boltzmann failed to recognize or, at least, to acknowledge. This was the implicit assumption that each microstate is equally probable - in other words, that as a system evolves in accordance with the fundamental laws of molecular mechanics, it will, on average, spend an amount of time in each macrostate that is proportional to the number of microstates in the macrostate. To restate the matter using more technical terminology, the assumption is that the fraction of time the system spends in each macrostate is proportional to that macrostate's "volume" expressed as a fraction of the total accessible volume of the "phase space" for the system. (A "phase space" is an abstract multidimensional space representing all possible microstates of a physical system.) That assumption, in turn, is a consequence of what is now generally referred to as the "ergodic hypothesis." The hypothesis had been an explicitly-acknowledged or implicitly-assumed element of Boltzmann's reasoning about the statistical behavior of molecules from the first time he articulated it in a series of papers in the 1860s. But - despite several attempts - Boltzmann never offered any persuasive argument that real systems of molecules do display ergodic behavior, and at times he expressed doubt as to whether the hypothesis was even true. Ever since Boltzmann's 1877 tour de force, mathematicians and physicists have been attempting to justify the ergodic hypothesis - to show that it accurately describes the statistical behavior of molecules moving about in conformity with the fundamental laws of mechanics (whether classical or quantum). This is among the "set of issues that continue to plague the foundations of the theory [of statistical mechanics]" (Lawrence Sklar, "Physics and Chance: Philosophical Issues in the Foundations of Statistical Mechanics" (1993)). It was the problem that the mathematician John von Neumann tackled in the papers offered here. "Von Neumann [1903-1957] may have been the last representative of a once-flourishing and numerous group, the great mathematicians who were equally at home in pure and applied mathematics and who throughout their careers maintained a steady production in both directions. . [I]n a profession where quick minds are somewhat commonplace, his amazing rapidity was proverbial. There is hardly a single important part of the mathematics of the 1930's with which he had not at least a passing acquaintance, and the same is probably true of theoretical physics." (DSB). His important contributions to pure and applied mathematics include the theory of Hilbert spaces, his axiomatization of quantum mechanics, and his pioneering work in game theory, in quantum statistics and thermodynamics, in numerical analysis, in the theory of computer architecture, in the theory of automata, and in game theory. Essentially, what von Neumann demonstrated in the offered papers is that the ergodic hypothesis is equivalent to an assumption called "metric indecomposability," or "metric transitivity." In simple, intuitive terms, and omitting some technical qualifications, a physical system is metrically indecomposable (or transitive) if it evolves in such a way that regardless of its initial microstate, it will not remain confined within any sub-region of the system's phase space. Von Neumann showed that a metrically transitive system will have properties whose average over time is equal to the average value of those properties over the phase space of all possible microstates. And that, in turn, is sufficient to underwrite Boltzmann's conclusion that the equilibrium macrostate, representing the overwhelmingly largest percentage of a phase space, is ipso facto the most probable macrostate. Von Neumann communicated his results to George Birkhoff prior to publication, and Birkhoff managed to prove a sharper, stronger version of von Neumann's theorem (the "pointwise" ergodic theorem v. von Neumann's "mean" ergodic theorem), which, through the vagaries of the publication process, appeared before von Neumann's paper. In March 1931, two months after von Neumann's proof was published, von Neumann and Birkhoff published two separate papers in Proc.

Taschenbuch. Condition: Neu. Druck auf Anfrage Neuware - Printed after ordering - Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Taschenbuch. Condition: Neu. Einführung in die Geophysik | III Dynamische Ozeanographie | A. Defant | Taschenbuch | x | Deutsch | Springer Vienna | EAN 9783709156490 | Verantwortliche Person für die EU: Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg, juergen[dot]hartmann[at]springer[dot]com | Anbieter: preigu.

Condition: New. Print on Demand pp. 236 67:B&W 6.69 x 9.61 in or 244 x 170 mm (Pinched Crown) Perfect Bound on White w/Gloss Lam.

Taschenbuch. Condition: Neu. This item is printed on demand - it takes 3-4 days longer - Neuware -Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben. 236 pp. Deutsch.

Condition: New. PRINT ON DEMAND pp. 236.

Condition: New. Dieser Artikel ist ein Print on Demand Artikel und wird nach Ihrer Bestellung fuer Sie gedruckt. Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfaengen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen fuer die historische wie auch die disziplingeschichtliche.

Taschenbuch. Condition: Neu. This item is printed on demand - Print on Demand Titel. Neuware -Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.Springer Verlag GmbH, Tiergartenstr. 17, 69121 Heidelberg 236 pp. Deutsch.