Exact Energy Expansion of the two-dimensional Dyson Gas for Odd Values of $\Gamma/2$

TL;DR

This paper derives exact energy expansions for the two-dimensional Dyson gas at odd values of b3/2, providing new finite-size corrections, confirming known results at b3=2, and linking to random matrix theory for small systems.

Contribution

It introduces an exact method to compute the excess energy of the 2dOCP for odd b3/2 values, including finite-size corrections and connections to random matrix theory.

Findings

Exact energy expansion for odd b3/2 values of the Dyson gas.

Confirmation of known results at b3=2 using an alternative approach.

Analytical pair correlation function linked to Ginibre ensemble.

Abstract

Using the expansion on monomial functions of the Vandermonde determinant to the power $\Gamma=Q^2/(k_BT)$, a way to find the excess energy $U_{exc}$ of the two dimensional one component plasma 2dOCP on the hard and soft disk (or Dyson Gas) for odd values of $\Gamma/2$ is provided. At $\Gamma=2$, the current study not only corroborates the result for the particle-particle energy contribution of the Dyson gas found by Shakirov by using an alternative approach but also provides the exact $N$-finite expansion of the excess energy of the 2dOCP on the hard disk. The excess energy is fitted to an ansatz of the form $U_{exc} = K^1 N + K^2 \sqrt{N} + K^3 + K^4/N + O(1/N^2)$ to study the finite-size corrections with $K^i$ coefficients and $N$ the number of particles. In particular, the bulk term of the excess energy is in agreement with the well known result of Jancovici for the hard disk in the thermodynamic limit. Finally, an expression is found for the pair correlation function which still keeps a link with the random matrix theory via the kernel in the Ginibre Ensemble for odd values of $\Gamma/2$. A comparison between analytical 2-body density function and histograms obtained with Monte Carlo simulations for small systems and $\Gamma=2,6,10,\ldots$ shows that the approach described in this work may be used to study analytically the crossover behaviour from a disordered system to small crystals. Key words: Coulomb gas, one-component plasma, Ginibre ensemble, solvable models