On the Foundations of Kinetic Theory

Abstract

We discuss the problem of deriving an 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 a suitable positivity preserving operator Λ such that the unitary group U_t induced from dynamics satisfies the intertwining relation

$$ \Lambda {{\text{U}}_{\text{t}}} = {\text{W}}_{\text{t}}^*\Lambda {\text{ }},{\text{ t }}_ = ^ > {\text{0}} $$

with the contraction semigroup W_t of a strongly irreversible stochastic Markov process. Two cases are of special interest: i) Λ = P is a projection operator, ii) Λ has a densely defined inverse. 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 _t ln ρ _t dµ with ρ _t denoting the coarse-grained distribution with respect to a K partition obtained from p_t ≡ U_tp 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 operator or the Λ transformation are dynamics dependent and break the time reversal.