The Annals of Applied Probability

Phase transitions in the one-dimensional Coulomb gas ensembles

Tatyana S. Turova

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We consider the system of particles on a finite interval with pairwise nearest neighbours interaction and external force. This model was introduced by Malyshev [Probl. Inf. Transm. 51 (2015) 31–36] to study the flow of charged particles on a rigorous mathematical level. It is a simplified version of a 3-dimensional classical Coulomb gas model. We study Gibbs distribution at finite positive temperature extending recent results on the zero temperature case (ground states). We derive the asymptotics for the mean and for the variances of the distances between the neighbouring charges. We prove that depending on the strength of the external force there are several phase transitions in the local structure of the configuration of the particles in the limit when the number of particles goes to infinity. We identify 5 different phases for any positive temperature.

The proofs rely on a conditional central limit theorem for nonidentical random variables, which has an interest on its own.

Article information

Ann. Appl. Probab., Volume 28, Number 2 (2018), 1249-1291.

Received: December 2016
Revised: July 2017
First available in Project Euclid: 11 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82B21: Continuum models (systems of particles, etc.) 82B26: Phase transitions (general) 60F05: Central limit and other weak theorems

Coulomb gas phase transitions Gibbs ensemble


Turova, Tatyana S. Phase transitions in the one-dimensional Coulomb gas ensembles. Ann. Appl. Probab. 28 (2018), no. 2, 1249--1291. doi:10.1214/17-AAP1329.

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