  An information theory description of finite systems explicitly evolving in
time is presented for classical as well as quantum mechanics. We impose a
variational principle on the Shannon entropy at a given time while the
constraints are set at a former time. The resulting density matrix deviates
from the Boltzmann kernel and contains explicit time odd components which can
be interpreted as collective flows. Applications include quantum brownian
motion, linear response theory, out of equilibrium situations for which the
relevant information is collected within different time scales before entropy
saturation, and the dynamics of the expansion.
