# Gaussian Random Matrix Ensembles in Phase Space

## TL;DR

This paper introduces a new class of Gaussian random matrix ensembles in phase space, extending classical ensembles and deriving thermodynamic properties for these ensembles with applications to gases and quantum matrices.

## Contribution

It proposes a novel class of Gaussian random matrix ensembles in phase space, extending classical ensembles and deriving their thermodynamic properties.

## Key findings

- Derived thermodynamic magnitudes for the new ensembles.
- Provided examples with nonideal and ideal gases.
- Established distribution functions from maximum entropy principle.

## Abstract

A new class of Random Matrix Ensembles is introduced. The Gaussian orthogonal, unitary, and symplectic ensembles GOE, GUE, and GSE, of random matrices are analogous to the classical Gibbs ensemble governed by Boltzmann's distribution in the coordinate space. The proposed new class of Random Matrix ensembles is an extension of the above Gaussian ensembles and it is analogous to the canonical Gibbs ensemble governed by Maxwell-Boltzmann's distribution in phase space. The thermodynamical magnitudes of partition function, intrinsic energy, free energy of Helmholtz, free energy of Gibbs, enthalpy, as well as entropy, equation of state, and heat capacities, are derived for the new ensemble. The examples of nonideal gas with quadratic potential energy as well as ideal gas of quantum matrices are provided. The distribution function for the new ensembles is derived from the maximum entropy principle.