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Full-Text Articles in Physical Sciences and Mathematics
Global Optimization, The Gaussian Ensemble, And Universal Ensemble Equivalence., M Costeniuc, Rs Ellis, H Touchette, B Turkington
Global Optimization, The Gaussian Ensemble, And Universal Ensemble Equivalence., M Costeniuc, Rs Ellis, H Touchette, B Turkington
Mathematics and Statistics Department Faculty Publication Series
Given a constrained minimization problem, under what conditions does there exist a related, unconstrained
problem having the same minimum points? This basic question in global optimization
motivates this paper, which answers it from the viewpoint of statistical mechanics. In this context, it
reduces to the fundamental question of the equivalence and nonequivalence of ensembles, which is
analyzed using the theory of large deviations and the theory of convex functions.
In a 2000 paper appearing in the Journal of Statistical Physics, we gave necessary and sufficient
conditions for ensemble equivalence and nonequivalence in terms of support and concavity
properties of the …
The Generalized Canonical Ensemble And Its Universal Equivalence With The Microcanonical Ensemble, M Costeniuc, Rs Ellis, H Touchette, B Turkington
The Generalized Canonical Ensemble And Its Universal Equivalence With The Microcanonical Ensemble, M Costeniuc, Rs Ellis, H Touchette, B Turkington
Mathematics and Statistics Department Faculty Publication Series
This paper shows for a general class of statistical mechanical models that when the microcanonical and canonical ensembles are nonequivalent on a subset of values of the energy, there often exists a generalized canonical ensemble that satisfies a strong form of equivalence with the microcanonical ensemble that we call universal equivalence. The generalized canonical ensemble that we consider is obtained from the standard canonical ensemble by adding an exponential factor involving a continuous function g of the Hamiltonian. For example, if the microcanonical entropy is C2, then universal equivalence of ensembles holds with g taken from a class …
Analysis Of Phase Transitions In The Mean-Field Blume-Emery-Griffiths Model, Rs Ellis, Pt Otto, H Touchette
Analysis Of Phase Transitions In The Mean-Field Blume-Emery-Griffiths Model, Rs Ellis, Pt Otto, H Touchette
Mathematics and Statistics Department Faculty Publication Series
In this paper we give a complete analysis of the phase transitions in the mean-field Blume-Emery-Griffiths lattice-spin model with respect to the canonical ensemble, showing both a second-order, continuous phase transition and a first-order, discontinuous phase transition for appropriate values of the thermodynamic parameters that define the model. These phase transitions are analyzed both in terms of the empirical measure and the spin per site by studying bifurcation phenomena of the corresponding sets of canonical equilibrium macrostates, which are defined via large deviation principles. Analogous phase transitions with respect to the microcanonical ensemble are also studied via a combination of …
Large Deviation Principles And Complete Equivalence And Nonequivalence Results For Pure And Mixed Ensembles, Rs Ellis, K Haven, B Turkington
Large Deviation Principles And Complete Equivalence And Nonequivalence Results For Pure And Mixed Ensembles, Rs Ellis, K Haven, B Turkington
Mathematics and Statistics Department Faculty Publication Series
We consider a general class of statistical mechanical models of coherent structures in turbulence, which includes models of two-dimensional fluid motion, quasi-geostrophic flows, and dispersive waves. First, large deviation principles are proved for the canonical ensemble and the microcanonical ensemble. For each ensemble the set of equilibrium macrostates is defined as the set on which the corresponding rate function attains its minimum of 0. We then present complete equivalence and nonequivalence results at the level of equilibrium macrostates for the two ensembles. Microcanonical equilibrium macrostates are characterized as the solutions of a certain constrained minimization problem, while canonical equilibrium macrostates …