  In this paper we review various information-theoretic characterizations of
the approach to equilibrium in biological systems. The replicator equation,
evolutionary game theory, Markov processes and chemical reaction networks all
describe the dynamics of a population or probability distribution. Under
suitable assumptions, the distribution will approach an equilibrium with the
passage of time. Relative entropy - that is, the Kullback--Leibler divergence,
or various generalizations of this - provides a quantitative measure of how far
from equilibrium the system is. We explain various theorems that give
conditions under which relative entropy is nonincreasing. In biochemical
applications these results can be seen as versions of the Second Law of
Thermodynamics, stating that free energy can never increase with the passage of
time. In ecological applications, they make precise the notion that a
population gains information from its environment as it approaches equilibrium.
