High temperature dimensional reduction in Snyder space

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.