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MaxEnt inference algorithm and information theory are relevant for the timeevolution of macroscopic systems considered as problem of incompleteinformation. Two different MaxEnt approaches are introduced in this work, bothapplied to prediction of time evolution for closed Hamiltonian systems. Thefirst one is based on Liouville equation for the conditional probabilitydistribution, introduced as a strict microscopic constraint on time evolutionin phase space. The conditional probability distribution is defined for the setof microstates associated with the set of phase space paths determined bysolutions of Hamilton's equations. The MaxEnt inference algorithm withShannon's concept of the conditional information entropy is then applied toprediction, consistently with this strict microscopic constraint on timeevolution in phase space. The second approach is based on the same concepts,with a difference that Liouville equation for the conditional probabilitydistribution is introduced as a macroscopic constraint given by a phase spaceaverage. We consider the incomplete nature of our information about microscopicdynamics in a rational way that is consistent with Jaynes' formulation ofpredictive statistical mechanics. Maximization of the conditional informationentropy subject to this macroscopic constraint leads to a loss of correlationbetween the initial phase space paths and final microstates. Informationentropy 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 lossof statistical correlation. In accordance with Jaynes, irreversibility appearsas a consequence of gradual loss of information about possible microstates ofthe system.

Macroscopic time evolution and MaxEnt inference for closed systems with Hamiltonian dynamics