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Statistical mechanics of classical particles with logarithmic interactions. (English) Zbl 0811.76002

The inhomogeneous mean-field thermodynamic limit is evaluated for both the canonical thermodynamic functions and the states of classical two- dimensional systems with logarithmically interacting point-like particles. The results are applicable to various physical models of translation invariant plasmas, gravitating systems and to planar fluid vortex motion. For attractive interactions there exists a critical behaviour which can be interpreted as an extreme case of a second-order phase transition. To include these interactions, a new inequality for configurational integrals is derived from the arithmetic-geometric mean inequality. The method developed in this paper is applicable to systems with fairly general interactions in all space dimensions. It also yields a new proof of the Trudinger-Moser inequality known in differential geometry.

MSC:

76A02 Foundations of fluid mechanics
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
70F99 Dynamics of a system of particles, including celestial mechanics
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