Bose - Einstein condensation in arbitrary dimensions

TL;DR

This paper generalizes Bose-Einstein condensation to arbitrary dimensions, analyzing how the condensation temperature varies with dimensionality and revealing a finite limit as dimensions approach infinity.

Contribution

It derives the density of states and condensation temperature for bosons in arbitrary dimensions, extending the understanding of BEC beyond three dimensions.

Findings

Condensation occurs in dimensions d ≥ 3.

Condensation temperature drops sharply above three dimensions.

In infinite dimensions, the temperature approaches a finite value.

Abstract

The density of bosonic states are calculated for spinless free massive bosons in generalised d dimensions. The number of bosons are calculated in the lowest energy state. The Bose Einstein condensation was found in generalised d dimensions (on and above d = 3) and the condensation temperature is calculated which is observed to drop abruptly above three dimensions and decreases monotonically as the dimensionalities of the system increases. The rate of fall of the condensation temperature decreases as the dimensionality increases. Interestingly, in the limit of infinite dimensions, the condensation temperature is observed to approach a nonzero finite value.