MATHPH In the literature: open

Condensation on all length scales in the dilute Bose gas

In plain words

In a very dilute gas the particles rarely meet, so condensation should be easiest to prove there. Current proofs work only in regions somewhat larger than the healing length, the distance over which the condensate recovers from a disturbance.

Precise statement

Dilute regime $\rho a^3 << 1$ ($a = \text{scattering length of }v$). Show that the condensate fraction in a box of side $L$ tends to $1 - O((\rho a^3)^{1/2})$ uniformly as $L \to \infty$, extending results valid for $L$ up to a power of the healing length $(8 \pi \rho a)^{-1/2}$ times a negative power of $\rho a^3$. Answer: a proof.

What would settle it

A proof of a condensate fraction near 1 uniform in $L$ for fixed small $\rho a^{3}$.

Status in the literature

Fournais (2021) proved condensation on length scales larger than the healing length; the infinite-volume statement is open.