Proof of the Ergodic Theorem. WITH: Proof of the Quasi-Ergodic Hypothesis. WITH: Physical Applications of the Ergodic Hypothesis. WITH: Recent Contributions to the Ergodic Theory

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.…