Bose-Einstein condensation in interacting gases

5 problems

• Condensation of a repulsive Bose gas in the thermodynamic limit

  Open

  Prove that a gas of mutually repelling bosons at low temperature puts a finite fraction of its particles into one quantum state when the container is infinitely large at fixed density. Without interactions this is a textbook calculation; with them there is no accepted proof.

• Condensation on all length scales in the dilute Bose gas

  Open

  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.

• Condensation of lattice hard-core bosons away from half filling

  Open

  For bosons hopping on a 3D lattice with at most one particle per site, condensation is proven only when exactly half the sites are filled, because the proof uses a mirror symmetry that holds only there. At any other filling it is unproven.

• Shift of the condensation temperature caused by weak repulsion

  Open

  Weak repulsion between bosons raises their condensation temperature slightly, in proportion to the interaction strength. The coefficient is known only from computer simulations.

• Third-order term in the dilute Bose gas ground-state energy

  Partially resolved

  The energy of a dilute repulsive Bose gas is known rigorously to two terms in powers of the density. The next predicted term, with a logarithm, is proven only from above.