[Paper Review] The Coulomb gas, potential theory and phase transitions

Robert J. Berman|arXiv (Cornell University)|Nov 26, 2018

Geometry and complex manifolds41 references3 citations

TL;DR

This paper provides a potential-theoretic characterization of measures for which the Coulomb gas and more general Riesz gases exhibit well-behaved statistical mechanics, ensuring a continuous transition to zero temperature via Gamma-convergence of large deviation principles. It proves that the Hausdorff measure on a Lipschitz hypersurface avoids zeroth-order phase transitions, while constructing explicit examples of absolutely continuous measures on the plane that do exhibit such transitions, linking the phenomenon to orthogonal polynomials and Bernstein-Markov inequalities.

ABSTRACT

We give a potential-theoretic characterization of measures which have the property that the corresponding Coulomb gas is well-behaved and similarly for more general Riesz gases. This means that the laws of the empirical measures of the corresponding random point process satisfy a Large Deviation Principle with a rate functional which depends continuously on the temperature, in the sense of Gamma-convergence. Equivalently, there is no zeroth-order phase transition at zero temperature. This is shown to be the case for the Hausdorff measure on a Lipschitz hypersurface. We also provide explicit examples of measures which are absolutely continuous with respect to Lesbesgue measure, such that the corresponding 2d Coulomb exhibits a zeroth-order phase transition. This is based on relations to Ullman's criterion in the theory of orthogonal polynomials and Bernstein-Markov inequalities.

Motivation & Objective

• To characterize measures for which the Coulomb gas has a well-behaved thermodynamic limit without phase transitions at zero temperature.
• To establish conditions under which the large deviation principle for empirical measures Gamma-converges as temperature approaches zero.
• To identify explicit examples of absolutely continuous measures on R² that induce zeroth-order phase transitions in 2D Coulomb gases.
• To connect the existence of such phase transitions to classical results in orthogonal polynomial theory, particularly Ullman's criterion and Bernstein-Markov inequalities.

Proposed method

• Uses potential-theoretic tools to analyze equilibrium measures and energy functionals in Coulomb and Riesz gas models.
• Applies Gamma-convergence to the rate functional of the large deviation principle as temperature tends to zero.
• Characterizes the absence of zeroth-order phase transitions via continuity of the rate functional in temperature.
• Employs Ullman's criterion for extremal measures in orthogonal polynomial theory to construct counterexamples with phase transitions.
• Analyzes the interplay between the geometry of the support and the density of the reference measure to determine phase behavior.
• Leverages Bernstein-Markov inequalities to establish conditions under which the equilibrium measure fails to converge continuously to the zero-temperature limit.

Experimental results

Research questions

• RQ1 Under what conditions on the reference measure does the Coulomb gas exhibit a continuous transition to zero temperature, avoiding zeroth-order phase transitions?
• RQ2 How does the Hausdorff measure on a Lipschitz hypersurface behave in the context of large deviation principles for Coulomb gases?
• RQ3 Can explicit examples of absolutely continuous measures on R² be constructed such that the corresponding 2D Coulomb gas exhibits a zeroth-order phase transition?
• RQ4 What is the connection between the existence of such phase transitions and classical results in orthogonal polynomial theory, such as Ullman's criterion?
• RQ5 How do Bernstein-Markov inequalities constrain the behavior of equilibrium measures in the zero-temperature limit?

Key findings

• The Hausdorff measure on a Lipschitz hypersurface yields a Coulomb gas without zeroth-order phase transitions, as the rate functional Gamma-converges continuously to zero temperature.
• Explicit examples of absolutely continuous measures on R² exist for which the 2D Coulomb gas exhibits a zeroth-order phase transition, demonstrating discontinuity in the zero-temperature limit.
• The presence of such phase transitions is linked to violations of Ullman's criterion in orthogonal polynomial theory, indicating a deep connection between statistical mechanics and approximation theory.
• Bernstein-Markov inequalities are shown to be instrumental in determining whether the equilibrium measure converges continuously to the minimizer of the energy at zero temperature.
• The paper establishes that the absence of phase transitions is equivalent to the continuity of the rate functional in temperature under Gamma-convergence.
• The analysis reveals that the geometry and density of the reference measure critically influence the thermodynamic stability of Coulomb gases.