Concept of entropy for nonequilibrium states of closed many-body systems

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Abstract
In this paper we study three fundamental questions: the characterization of the nonequilibrium states of a macroscopic system using exclusively the information that is controllable by the experimenter through the constraints imposed on the system, the construction of a variational method generating a probability distribution function in which the information available to the experimenter is contained, and finally the study of the time behavior of a functional, here referred to as the Shannon-Jaynes entropy, in order to examine the existence of a criterion for irreversibility. It is shown that the answer to the first two questions is provided by a theorem establishing the equivalence between the coarse-graining operation in phase space as suggested by Ehrenfest and the action of a projection operator, in fact, Zwanzig’s projector, acting on the full probability distribution function containing all the information available when the system is initially prepared. The third question is partially answered by deriving an inequality characterizing the fact that every time the system is observed, information is lost. Yet the full proof establishing the equivalent of Boltzmann’s H theorem is only qualitatively analyzed. The connection between this work and a similar one carried out from a macroscopic point of view is also established.