The entropy of dense non-commutative fermion gases

TL;DR

This paper analyzes the thermodynamic properties of non-commutative fermion gases in two and three dimensions, revealing non-extensive entropy, incompressibility in 2D, and the importance of rotational symmetry for 3D behavior.

Contribution

It provides analytic expressions for entropy and pressure in non-commutative fermion gases, highlighting the role of symmetry and system size in their thermodynamic behavior.

Findings

Entropy is non-extensive and proportional to the square root of system size in 2D.

Incompressibility is observed in 2D but not in 3D due to symmetry constraints.

Restoring rotational symmetry may induce incompressibility in 3D systems.

Abstract

We investigate the properties of two- and three-dimensional non-commutative fermion gases with fixed total z-component of angular momentum, J_z, and at high density for the simplest form of non-commutativity involving constant spatial commutators. Analytic expressions for the entropy and pressure are found. The entropy exhibits non-extensive behaviour while the pressure reveals the presence of incompressibility in two, but not in three dimensions. Remarkably, for two-dimensional systems close to the incompressible density, the entropy is proportional to the square root of the system size, i.e., for such systems the number of microscopic degrees of freedom is determined by the circumference, rather than the area (size) of the system. The absence of incompressibility in three dimensions, and subsequently also the absence of a scaling law for the entropy analogous to the one found in two dimensions, is attributed to the form of the non-commutativity used here, the breaking of the rotational symmetry it implies and the subsequent constraint on J_z, rather than the angular momentum J. Restoring the rotational symmetry while constraining the total angular momentum J seems to be crucial for incompressibility in three dimensions. We briefly discuss ways in which this may be done and point out possible obstacles.