  MaxEnt inference algorithm and information theory are relevant for the time
evolution of macroscopic systems considered as problem of incomplete
information. Two different MaxEnt approaches are introduced in this work, both
applied to prediction of time evolution for closed Hamiltonian systems. The
first one is based on Liouville equation for the conditional probability
distribution, introduced as a strict microscopic constraint on time evolution
in phase space. The conditional probability distribution is defined for the set
of microstates associated with the set of phase space paths determined by
solutions of Hamilton's equations. The MaxEnt inference algorithm with
Shannon's concept of the conditional information entropy is then applied to
prediction, consistently with this strict microscopic constraint on time
evolution in phase space. The second approach is based on the same concepts,
with a difference that Liouville equation for the conditional probability
distribution is introduced as a macroscopic constraint given by a phase space
average. We consider the incomplete nature of our information about microscopic
dynamics in a rational way that is consistent with Jaynes' formulation of
predictive statistical mechanics. Maximization of the conditional information
entropy subject to this macroscopic constraint leads to a loss of correlation
between the initial phase space paths and final microstates. Information
entropy is the theoretic upper bound on the conditional information entropy,
with the upper bound attained only in case of the complete loss of correlation.
In this alternative approach to prediction of macroscopic time evolution,
maximization of the conditional information entropy is equivalent to the loss
of statistical correlation. In accordance with Jaynes, irreversibility appears
as a consequence of gradual loss of information about possible microstates of
the system.
