How to measure the free energy and partition function from atom-atom correlations

We propose an experimental approach for determining thermodynamic properties of ultracold atomic gases with short-range interactions. As a test case, we focus on the one-dimensional (1D) Bose gas described by the integrable Lieb-Liniger model. The proposed approach relies on deducing the Helmholtz or Landau free energy directly from measurements of local atom-atom correlations by utilising the inversion of a finite-temperature version of the Hellmann-Feynman theorem. We demonstrate this approach theoretically by deriving approximate analytic expressions for the free energies in specific asymptotic regimes of the 1D Bose gas and find excellent agreement with the exact results based on the thermodynamic Bethe ansatz available for this integrable model.