  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.
