Correlations, operations and the second law of thermodynamics

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.