On Nonunitary Equivalence Between Unitary Group of Dynamics and Contraction Semigroups of Markov Processes

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 suitable positivity preserving operator ∧ such that the unitary group U_t induced from dynamics satisfies the intertwining relation:

$$ \wedge {U_{t}} = W_{t}^{*} \wedge $$

with the contraction semigroup W ^*_t of a strongly irreversible stochastic Markov process. Two cases are of special interest: (i) ∧ is a projection operator, (ii) ∧ has densely defined inverse ∧⁻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 relatin exists is identical with the class of K-flows or K-systems. As a corollary of our consideration it follows that the function $\int {{{\hat{\rho }}_{t}}\ell n{{\hat{\rho }}_{t}}d\mu }$ with ${\hat{\rho }_{t}}$ 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 it is the operation of coarse-graining with respect to K partition that breaks the symmetry.