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Almost every quantum system interacts with a large environment, so the exactquantum mechanical description of its evolution is impossible. One has toresort to approximate description, usually in the form of a master equation.There are at least two basic requirements for such a description: first, itshould preserve the positivity of probabilities; second, it should correctlydescribe the equilibration process for systems coupled to a single thermalbath. Existing two widespread descriptions of evolution fail to satisfy atleast one of those conditions. The so-called Davies master equation, whilepreserving the positivity of probabilities, fails to describe thermalizationproperly. On the other hand, the Bloch-Redfield master equation violates thefirst condition, but it correctly describes equilibration, at least foroff-diagonal elements for several important scenarios. However, is it possibleto have a description of open system dynamics that would share both features?In this paper, we partially resolve this problem in the weak-coupling limit:(i) We provide a general form of the proper thermal equilibrium state (theso-called mean-force state) for an arbitrary open system. (ii) We provide thesolution for the steady-state coherences for a whole class of master equations,and in particular, we show that the solution coincides with the mean-forceHamiltonian for the Bloch-Redfield equation. (iii) We consider the cumulantequation, which is explicitly completely positive, and we show that itssteady-state coherences are the same as one of the Bloch-Redfield dynamics (andthe mean-force state accordingly). (iv) We solve the correction to the diagonalpart of the stationary state for a two-level system both for the Bloch-Redfieldand cumulant equation, showing that the solution of the cumulant is very closeto the mean-force state, whereas the Bloch-Redfield differs significantly.

Towards reconciliation of completely positive open system dynamics with the equilibration postulate