Condensed matter physics
Bose–Einstein condensation: theory
Bose–Einstein condensation is macroscopic occupation of the ground state by bosons at low temperature. For a uniform ideal 3D gas, the critical temperature follows from the density of states; weak interactions, finite size, and trapping modify the threshold and transition.
At sufficiently low temperature, the excited states have a finite capacity. When the total particle number exceeds it, the excess occupies the same ground state. In an interacting system the condensate couples to the noncondensed cloud; it is not simply “all atoms at rest.”
Definition: Ideal-gas critical temperature
This applies to a uniform ideal three-dimensional Bose gas of density n=N/V and mass m in the thermodynamic limit. ζ(3/2)≈2.612. At threshold μ approaches the bottom of the spectrum; below it the ground state has macroscopic occupation.
From state counting to condensate
The number of excited particles follows by integrating the density of states times the Bose occupation. In uniform 3D, this integral reaches its maximum when μ is at the spectral bottom, yielding Tc. Below threshold the ideal condensate fraction is N₀/N=1−(T/Tc)^(3/2).
Experiments use cold atoms in traps with weak interactions. The measured critical temperature can differ from the uniform-gas prediction because of inhomogeneous density, finite particle number, and interactions; comparisons must specify the model.
Quick check
In a dilute, weakly interacting gas, short-range elastic scattering is characterized by the scattering length a_s. Near zero temperature the condensate density n₀ sets an interaction scale gn₀, with g=4πℏ²a_s/m; collective excitations become phonons at long wavelengths. An interacting condensate can therefore be superfluid under suitable conditions, but single-particle condensation and superfluidity are distinct concepts. This description assumes a dilute gas, na_s³≪1.
The Gross–Pitaevskii equation describes the condensate field at mean-field level, balancing the external potential and contact interaction against quantum kinetic energy. The healing length ξ≈ℏ/√(2mgn₀) sets the scale over which density recovers from a disturbance; a vortex has a core of order ξ and quantized circulation. Near the transition, thermal fluctuations and beyond-mean-field correlations matter, so this approximation is no longer sufficient.
Below Tc, what is the ideal condensate fraction in a uniform 3D Bose gas?
Which sign is in the denominator of the corresponding distribution?
References
- Landau, Lifshitz (1980). Statistical Physics
- Charles Kittel, Herbert Kroemer (1980). Thermal Physics