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Journal of Statistical Physics

Foundations of the quantum statistics of systems in equilibrium: A measuretheoretic approach

Garrison Sposito

Published Apr 1, 1974• 3 citations
Publisher
DOI
10.1007/bf01012254

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氏名 池田 達彦The main theme of this thesis is the foundation of statistical mechanics in isolated quantum systems: how equilibrium states, or macroscopically unchanging states, emerge in the course of the unitary time evolution, why they are well described by the microcanonical ensemble, and how the thermodynamic entropy is interpreted from a quantum-mechanical viewpoint.Isolated quantum systems are by definition not in contact with a heat bath and, thus, they never lose their memory of the initial state to reach thermal equilibrium by throwing away the information to the bath.Therefore, to address the questions raised above, we must find out the inherent ability of isolated quantum systems to undergo thermalization.The foundation has been studied since the early 20th century because an isolated system plays the most fundamental role in statistical mechanics.Presupposing that the microcanonical ensemble describes thermal equilibrium states in isolated systems, the canonical ensemble, for example, is deduced in systems in contact with a heat bath.Despite its importance, the foundation is still only presupposed and has not been fully proven yet.Recently, the theme has attracted renewed attention because isolated quantum systems have experimentally been realized using ultracold atoms, which are trapped by lasers in a high vacuum.Until then, it had been of purely theoretical interest, and thermalization of most quantum systems in reality had often been ascribed to the coupling with their environment.Now that thermalization is actually observed in ultracold atom experiments, the theme is of interest not only theoretically but also experimentally.
Statistical mechanics is often taken to be the paradigm of a successful inter-theoretic reduction, which explains the high-level phenomena by using the fundamental theories of physics together with some auxiliary hypotheses. The central idea of statistical mechanics is that in order to account for the thermodynamic regularities only partial information about the microstates is needed, concerning some macrovariables of them, that is, concerning the fact that they belong to certain macrostates in which all the microstates share the same macrovariables. An important example of a thermodynamic property, which needs to be explained in quantum statistical mechanics, is that of being in thermal equilibrium. The thermodynamic law of approach to equilibrium says that systems, prepared in non-equilibrium states, will invariably evolve to equilibrium states fixed by the constraints on these systems, and remain in them indefinitely.
We describe the large applicability of the Nonequilibrium Statistical Operator Method (NSOM) for the study of dissipative dynamic systems far from equilibrium. It is shown that the NSOM can be encompassed by a unifying variational principle, which produces a large family of NSO that contains existing examples as particular cases. Further, we review the application of the NSOM for the construction of a nonlinear quantum theory of large scope, and for the generation of a response function theory, for far-from-equilibrium Hamiltonian systems. An accompanying non-equilibrium thermodynamic Green's function theory is briefly described. Also it is shown that the NSOM provides mechano-statistical foundations for phenomenological irreversible thermodynamics, and for the important question of stability of far-from-equilibrium steady states and the emergence of self-organized dissipative structures in condensed matter.