We shall use two different perspectives to examine the minimization of information entropy on measures on the phase space of bounded domains, subject to constraints on the averages of the measures.
In Part I we shall adopt the approach of O. Lanford and A. Martin-Loef
to describe the set of all such constraints and will investigate how the set of constrains relates to the domain of the microcanonical thermodynamic limit entropy.
We shall then show that, for fixed constraints, the parameters of the corresponding grand canonical distribution converge, as volume increases, to the corresponding parameters (derivatives, when they exist) of the thermodynamic limit entropy.
In Part II we shall use a certain Orlicz manifold structure on the space of finite positive measures and a constrained optimization theorem for Banach manifolds to show that the critical points of the Gibbs entropy are grand canonical equilibria when the constraints are scalar, and local Giibs equilibria when the constraints are integrable functions. The thermodynamic parameters will now appear as Lagrange multipliers.
(Part I is joint work with S. Dostoglou and Jianfei Xue and has appeared in the Journal of Statistical Physics. Part II is joint with S. Dostoglou and under review.)
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