High temperature dimensional reduction in Snyder space

TL;DR

This paper explores the thermodynamics of an ideal gas in Snyder space, revealing a reduction in degrees of freedom at high temperatures that suggests an effective dimensional reduction from 3D to 1D at the Planck scale.

Contribution

It introduces a statistical mechanics framework in Snyder space and demonstrates a dimensional reduction at high energies due to minimal length effects.

Findings

Number of microstates reduces drastically at high energy

Degrees of freedom freeze at high temperature

Effective dimensional reduction from 3D to 1D at Planck scale

Abstract

In this paper, we formulate the statistical mechanics in Snyder space that supports the existence of a minimal length scale. We obtain the corresponding invariant Liouville volume which properly determines the number of microstates in the semiclassical regime. The results show that the number of accessible microstates drastically reduces at the high energy regime such that there is only one degree of freedom for a particle. Using the Liouville volume, we obtain the deformed partition function and we then study the thermodynamical properties of the ideal gas in this setup. Invoking the equipartition theorem, we show that $2/3$ of the degrees of freedom freeze at the high temperature regime when the thermal de Broglie wavelength becomes of the order of the Planck length. This reduction of the number of degrees of freedom suggests an effective dimensional reduction of the space from $3$ to $1$ at the Planck scale.