Relative entropy, interaction energy and the nature of dissipation
  Many thermodynamic relations involve inequalities, with equality if a process does not involve dissipation. In this article we provide equalities in which the dissipative contribution is shown to involve the relative entropy (a.k.a. Kullback-Leibler divergence). The processes considered are general time evolutions both in classical and quantum mechanics, and the initial state is sometimes thermal, sometimes partially so. By calculating a transport coefficient we show that indeed---at least in this case---the source of dissipation in that coefficient is the relative entropy. 
B. Gaveau, L. Granger, M. Moreau, and L. S. Schulman
cond-mat.stat-mech math-ph math.MP physics.chem-ph