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Antiferromagnetic Covariance Structure of Coulomb Chain

Tatyana S. Turova

Abstract

We consider a system of particles lined up on a finite interval with Coulomb 3-dimensional interactions between close neighbours, i.e. only a few other neighbours apart. This model was introduced by Malyshev (2015) to study the flow of charged particles. Notably even the nearest-neighbours interactions case, the only one studied previously, was proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Here we show that including interactions beyond the nearest-neighbours ones, leads to qualitatively new features. The order of covariances of distances between pairs of consecutive charges is changed when compared with the former nearest-neighbours case. Furthermore, we discover that the covariances between inter-spacings exhibit the antiferromagnetic property, namely they periodically change sign depending on the parity of number of spacings between them, while their amplitude decays.

Antiferromagnetic Covariance Structure of Coulomb Chain

Abstract

We consider a system of particles lined up on a finite interval with Coulomb 3-dimensional interactions between close neighbours, i.e. only a few other neighbours apart. This model was introduced by Malyshev (2015) to study the flow of charged particles. Notably even the nearest-neighbours interactions case, the only one studied previously, was proved to exhibit multiple phase transitions depending on the strength of the external force when the number of particles goes to infinity. Here we show that including interactions beyond the nearest-neighbours ones, leads to qualitatively new features. The order of covariances of distances between pairs of consecutive charges is changed when compared with the former nearest-neighbours case. Furthermore, we discover that the covariances between inter-spacings exhibit the antiferromagnetic property, namely they periodically change sign depending on the parity of number of spacings between them, while their amplitude decays.
Paper Structure (13 sections, 16 theorems, 371 equations)

This paper contains 13 sections, 16 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Models with Coulomb interactions
  3. Representation via conditional distribution
  4. Results
  5. Proofs
  6. Lagrange multiplier
  7. Exponential decay of correlations and approximation by a circular distribution
  8. Minima of $H_{\lambda}$ and $H_{\lambda}^{\circ}$
  9. Gaussian approximation
  10. Expectations and covariances
  11. Setting value for the Lagrange multiplier
  12. Central Limit Theorem
  13. Proof of Theorem \ref{['T1']}

Key Result

Proposition 1.1

(MZ, T) The following distributional identity holds for all $N \geq 2$:

Theorems & Definitions (21)

  • Proposition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Theorem 2.7
  • ...and 11 more