Abstract
We generalize the notion of “ground states” in the Pirogov-Sinai theory of first order phase transitions at low temperatures, applicable to lattice systems with a finite number of periodic ground states to that of “restricted ensembles” with equal free energies. A restricted ensemble is a Gibbs ensemble, i.e. equilibrium probability measure, on a restricted set of configurations in the phase space of the system. When a restricted ensemble contains only one configuration it coincides with a ground state. In the more general case the entropy is also important.
An example of a system we can treat by our methods is theq-state Potts model where we prove that forq sufficiently large there exists a temperature at which the system coexists inq+1 phases;q-ordered phases are small modifications of theq perfectly ordered ground states and one disordered phase which is a modification of the restricted ensemble consisting of all “perfectly disordered” (neighboring sites must have different spins) configurations. The free energy thus consists entirely of energy in the firstq-restricted ensembles and of entropy in the last one.
Our main motivation for this work is to develop a rigorous theory for phase transitions in continuum fluids in which there is no symmetry between the phases, e.g. the liquid-vapour phase transition. The present work goes a certain way in that direction.
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Communicated by A. Jaffe
Supported in part by NSF Grant Nr DMR81-14726-02
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Bricmont, J., Kuroda, K. & Lebowitz, J.L. First order phase transitions in lattice and continuous systems: Extension of Pirogov-Sinai theory. Commun.Math. Phys. 101, 501–538 (1985). https://doi.org/10.1007/BF01210743
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DOI: https://doi.org/10.1007/BF01210743