We define and characterize the ``chained'' Kullback-Leibler divergence $min_w D(p||w) + D(w||q)$ minimized over all intermediate distributions $w$, the analogous $k$-fold chained K-L divergence $min D(p||w_1) + D(w_1||w_2) + ldots + D(w_k||q)$ minimized over the entire path $(w_1,ldots,w_k)$, and analogous chained mutual informations.  These quantities arise in a large deviations analysis of a Markov chain on the set of types -- the Wright-Fisher model of neutral genetic drift: a population with allele distribution $q$ produces offspring with allele distribution $w$, which then produce offspring with allele distribution $p$, and so on.  We further define, chracterize, and efficiently compute a chained mutual information,

The chained divergences enjoy some of the same properties as the K-L divergence and appear in $k$-step versions of some of the same settings (like information projections and a conditional limit theorem).  We further characterize the optimal $k$-step ``path'' of distributions appearing in the definition, and find the limiting path as $k rightarrow infty$ to be a geodesic in the Fisher information metric, and a geometric interpretation of the limiting chained mutual information.  Finally we offer a thermodynamic interpretation of the chained divergence as the rate of operation of an appropriately defined Maxwell's demon.