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Critical temperatures and collapsing of two-dimensional Log gases

Rolf Andreasson, Ludvig Svensson

Abstract

We consider the canonical ensemble of a system of point particles on the sphere interacting via a logarithmic pair potential. In this setting, we study the associated Gibbs measure and partition function, and we derive explicit formulas relating the critical temperature, at which the partition function diverges, to a certain discrete optimization problem. We further show that the asymptotic behavior of both the partition function and the Gibbs measure near the critical temperature is governed by the same optimization problem. Our approach relies on the Fulton--MacPherson compactification of configuration spaces and analytic continuation of complex powers. To illustrate the results, we apply them to well-studied systems, including the two-component plasma and the Onsager model of turbulence. In particular, for the two-component plasma with general charges, we describe the formation of dipoles close to the critical temperature, which we determine explicitly.

Critical temperatures and collapsing of two-dimensional Log gases

Abstract

We consider the canonical ensemble of a system of point particles on the sphere interacting via a logarithmic pair potential. In this setting, we study the associated Gibbs measure and partition function, and we derive explicit formulas relating the critical temperature, at which the partition function diverges, to a certain discrete optimization problem. We further show that the asymptotic behavior of both the partition function and the Gibbs measure near the critical temperature is governed by the same optimization problem. Our approach relies on the Fulton--MacPherson compactification of configuration spaces and analytic continuation of complex powers. To illustrate the results, we apply them to well-studied systems, including the two-component plasma and the Onsager model of turbulence. In particular, for the two-component plasma with general charges, we describe the formation of dipoles close to the critical temperature, which we determine explicitly.

Paper Structure

This paper contains 26 sections, 14 theorems, 193 equations.

Table of Contents

  1. Introduction
  2. Comment on the proofs
  3. Applications
  4. Positive temperature
  5. Negative temperature
  6. Other geometries
  7. Outline of the Paper
  8. Preliminaries
  9. Complex geometry
  10. Divisors
  11. Blowups
  12. The relative canonical divisor
  13. Pulling back measures with analytic singularities
  14. Complex geometric formalism
  15. The Fulton--MacPherson compactification of configuration space
  16. ...and 11 more sections

Key Result

Theorem 1.3

The partition function $\mathcal{Z}(\beta)$ is finite precisely on the interval $(\beta^-,\beta^+)$ where and

Theorems & Definitions (33)

  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Example 2.1
  • Example 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 23 more