Thermodynamics on Fuzzy Spacetime

TL;DR

This paper explores how noncommutative fuzzy geometry affects thermodynamic properties of ideal gases, revealing unique behaviors in heat capacities and interparticle potentials compared to classical geometries, with implications for thermal reduction mechanisms.

Contribution

It provides a detailed analysis of thermodynamics on fuzzy spacetime, highlighting differences in heat capacities and interparticle interactions due to noncommutativity, including relativistic cases.

Findings

Heat capacities differ for bosons and fermions on fuzzy spheres.

Noncommutativity enhances statistical interparticle attraction or repulsion.

High-temperature limits lead to classical behavior and thermal reduction.

Abstract

We investigate the thermodynamics of non-relativistic and relativistic ideal gases on the spacetime with noncommutative fuzzy geometry. We first find that the heat capacities of the non-relativistic ideal boson and fermion on the fuzzy two-sphere have different values, contrast to that on the commutative geometry. We calculate the "statistical interparticle potential" therein and interprete this property as a result that the non-commutativity of the fuzzy sphere has an inclination to enhance the statistical "attraction (repulsion) interparticle potential" between boson (fermion). We also see that at high temperature the heat capacity approaches to zero. We next evaluate the heat capacities of the non-relativistic ideal boson and fermion on the product of the 1+D (with D=2,3) Minkowski spacetime by a fuzzy two-sphere and see that the fermion capacity could be a decreasing function of temperature in high-temperature limit, contrast to that always being an increasing function on the commutative geometry. Also, the boson and fermion heat capacities both approach to that on the 1+D Minkowski spacetime in high-temperature limit. We discuss these results and mention that the properties may be traced to the mechanism of "thermal reduction of the fuzzy space". We also investigate the same problems in the relativistic system with free Klein-Gordon field and Dirac field and find the similar properties.