Remarks on Shannon's Statistical Inference and the Second Law in Quantum Statistical Mechanics

概要

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We comment on a formulation of quantum statistical mechanics, which incorporates the statistical inference of Shannon. Our basic idea is to distinguish the dynamical entropy of von Neumann, H=-k Tr p ln p, in terms of the density matrix p(t), and the statistical amount of uncertainty of Shannon, S=-k Σ_n, p_n ln p_n, with p_n=<n∣p∣n> in the representation where the total energy and particle numbers are diagonal. These quantities satisfy the inequality S≥H. We propose to interprete Shannon's statistical inference as specifying the initial conditions of the system in terms of p_n.. A definition of macroscopic observables which are characterized by intrinsic time scales is given, and a quantum mechanical condition on the system, which ensures equilibrium, is discussed on the basis of time averaging. An interesting analogy of the change of entroy with the running coupling in renormalization group is noted. A salient feature of our approach is that the distinction between statistical aspects and dynamical aspects of quantum statistical mechanics is very transparent.
社団法人日本物理学会の論文
2002-01-15