Bose-Einstein condensation with a finite number of particles in a power-law trap

TL;DR

This paper derives an analytical expression for the Bose-Einstein condensation temperature in finite particle systems within power-law traps, revealing significant finite size effects and potential increases in critical temperature for higher-order potentials.

Contribution

It provides a new analytical framework for calculating the BEC transition temperature beyond the thermodynamic limit in power-law traps, including finite size corrections and effects of potential shape.

Findings

Finite size effects on $T_c$ are significant for small N.

In cubic traps, finite size effects cancel out.

Higher-order power-law traps can increase $T_c$ substantially.

Abstract

Bose-Einstein condensation (BEC) of an ideal gas is investigated, beyond the thermodynamic limit, for a finite number $N$ of particles trapped in a generic three-dimensional power-law potential. We derive an analytical expression for the condensation temperature $T_c$ in terms of a power series in $x_0=\epsilon_0/k_BT_c$, where $\epsilon_0$ denotes the zero-point energy of the trapping potential. This expression, which applies in cartesian, cylindrical and spherical power-law traps, is given analytically at infinite order. It is also given numerically for specific potential shapes as an expansion in powers of $x_0$ up to the second order. We show that, for a harmonic trap, the well known first order shift of the critical temperature $\Delta T_c/T_c \propto N^{-1/3}$ is inaccurate when $N \leqslant 10^{5}$, the next order (proportional to $N^{-1/2}$) being significant. We also show that finite size effects on the condensation temperature cancel out in a cubic trapping potential, e.g. $V(\mathbi{r}) \propto r^3$. Finally, we show that in a generic power-law potential of higher order, e.g. $V(\mathbi{r}) \propto r^\alpha$ with $\alpha > 3$, the shift of the critical temperature becomes positive. This effect provides a large increase of $T_c$ for relatively small atom numbers. For instance, an increase of about +40% is expected with $10^4$ atoms in a $V(\mathbi{r}) \propto r^{12}$ trapping potential.