Finite temperature correlations and density profiles of an inhomogeneous   interacting 1D Bose gas
  We calculate the density profiles and density correlation functions of the one-dimensional Bose gas in a harmonic trap, using the exact finite-temperature solutions for the uniform case, and applying a local density approximation. The results are valid for a trapping potential which is slowly varying relative to a correlation length. They allow a direct experimental test of the transition from the weak coupling Gross-Pitaevskii regime to the strong coupling, 'fermionic' Tonks-Girardeau regime. We also calculate the average two-particle correlation which characterizes the bulk properties of the sample, and find that it can be well approximated by the value of the local pair correlation in the trap center. 
K. V. Kheruntsyan, D. M. Gangardt, P. D. Drummond, and G. V.   Shlyapnikov