Macroscopic Time Evolution and MaxEnt Inference for Closed Systems with Hamiltonian Dynamics |
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Authors: | Domagoj Kui? Pa?ko ?upanovi? Davor Jureti? |
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Institution: | (1) Faculty of Science, University of Split, N. Tesle 12, 21000 Split, Croatia |
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Abstract: | 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, and the concept
of macroscopic reproducibility for time dependent processes. 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, and leads to corresponding loss
of information. In accordance with the original idea of Jaynes, irreversibility appears as a consequence of gradual loss of
information about possible microstates of the system. |
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