On the Foundations of Kinetic Theory

概要

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We discuss the problem of deriving exact Markovian master equation from dynamics without resorting to approximation schemes such as the weak coupling limit, Boltzmann-Grad limit etc. Mathematically, it is the problem of the existence of suitable positivity preserving operator Λ such that the unitary group U_t induced from dynamics satisfies the intertwining relation: ΛU_t=W_t^*Λ,t-__>0 with the contraction semigroup W_t of a strongly irreversible stochastic Markov process. Two cases are of special interest: (i) Λ is a projection operator, (ii) Λ has densely defined inverse Λ^<-1>. Our recent work which we summarize here, shows that the class of (classical) dynamical systems for which a suitable projection operator satisfying the above intertwining relation exists is identical with the class of K-flows or K-systems. As a corollary of our consideration it follows that the function ∫ρ^^^_tlnρ^^^_t with ρ^^^_t denoting the "coarse-grained" distribution with respect to a K-partition obtained from ρ_t≡U_tρ is a Boltzmann type H-function for K-flows. This is not in contradiction with the time reversal (velocity inversion) symmetry of dynamical evolution as the suitably constructed projection from K-partition is dynamics dependent and breaks the time-reversal symmetry explicitly.
理論物理学刊行会の論文
1981-06-10