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Shannon’s Statistical Inference and the H-theorem in Quantum Statistical Mechanics

By Kazuo Fujikawa


A formulation of quantum statistical mechanics, which incoporates the statistical inference of Shannon as its integral part, is discussed. Our basic idea is to distinguish the dynamical entropy of von Neumann in terms of the density matrix ˆρ(t), H = −kTrˆρln ˆρ, and the statistical amount of uncertainty of Shannon, S = −k ∑ n pn ln pn, with pn = (ˆρ)nn in the representation where the total energy and particle numbers are diagonal. We propose to interprete Shannon’s statistical inference as specifying the initial conditions of the system in terms of (ˆρ)nn, and thus prevent the statistical inference preceding physical dynamics. We argue that thermal equilibrium is ensured by a quantum counter part of mixing property, and that the physical entoropy of the final state, which is defined by a suitable time average in a themodynamic limit, is estimated by the maximum of Shannon’s S. In the context of the H-theorem in a broad sense, our picture is characterized as a specification of initial conditions by statistical inference on the basis of a limited amount of information available and a coarse-graining in the time direction. An interesting analogy with the renormalization group fixed point is also noted.

Year: 2000
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