  Completely positive trace preserving maps are essential for the formulation
of the second law of thermodynamics. The dynamics of quantum systems,
correlated with their environments, are in general not described by such maps.
We explore how this issue can be fixed by describing the classical analogue of
this problem. We consider correlated probability distributions, whose
subsequent system dynamics is ill-described by stochastic maps, and prescribe
the correct way to describe the dynamics. We use this prescription to discuss
the classical version of the second law, valid for correlated probability
distributions.
