(Received December 2, 1980)
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 Ut induced from dynamics satisfies the intertwining relation:
λ Ut = Wt* λ , (t ≥ 0)
with the contraction semigroup Wt 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 \int \hatρt ln \hatρt with pt denoting the “coarse-grained” distribution with respect to a K-partition obtained from ρt ≡ Ut ρ 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.
URL : http://ptp.ipap.jp/link?PTPS/69/101/
DOI : 10.1143/PTPS.69.101