Ideal Bose Gas

A Bose–Einstein condensate (BEC) is a state of matter of a dilute gas of bosons cooled to temperatures very close to 0K (usually ~100nK in experiments). Under such conditions, a large fraction of bosons occupies the lowest quantum state, at which point macroscopic quantum phenomena become apparent.

Overview

In this section, we summarize some basic and useful thermodynamic results for Bose-Einstein condensation in a uniform, non-interacting gas of bosons.

Thermodynamics of a Bose Gas

Phase Space Density

The fundamental difference between a BEC and a classical gas is the occupancy of a single-particle state. In a classical gas, the mean occupation number for a single quantum state satisfies the Boltzmann distribution $\langle n_{\nu }\rangle =e^{-(\epsilon _{\nu }-\mu )/kT}$ which is much less than unity. This feature is qualitatively captured by the ${\textit {Phase-spaceDensity}}$ defined as (3D, homogeneous gas)

$\rho _{D}=n\lambda _{T}^{3}\,,$

where $\lambda _{T}=(2\pi \hbar ^{2}/mkT)^{1/2}$ is the thermal de Broglie wavelength.

Some typical numbers for

• Classical gas
  • Atom density Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n \sim 10^{25} \m^{-3}}
  • Interatomic distance Failed to parse (unknown function "\nm"): {\displaystyle (1/n)^{1/3} \sim 3 \nm}
  • Thermal de Broglie wavelength Failed to parse (unknown function "\nm"): {\displaystyle \lambda_T \sim 10^{-2} \nm（T = 300K) }
  • $\rho _{D}\sim 10^{-8}$

The Bose Distribution

For non-interacting bosons in thermodynamic equilibrium, the mean occupation number of the single-particle state $\nu$ is

$\langle n_{\nu }\rangle ={\frac {1}{e^{(\epsilon _{\nu }-\mu )/kT}-1}}\,.$

. However, in a condesnate, the occupatyion number in the groud state is much larger than 1.