Solving 1D plasmas and 2D boundary problems using Jack polynomials and functional relations

The general one-dimensional log-sine gas is defined by restricting the positive and negative charges of a two-dimensional Coulomb gas to live on a circle. Depending on charge constrannts, this problem is equivalent to different boundary field theories.We study the electrically neutral case, which is equivalent to a two-dimensional free boson with an impurity cosine potential. We use two different methods: a perturbative one based on Jack symmetric functions, and a nonperturbative one based on the thermodynamic Bethe ansatz and functional relations. The first method allows us to find an explicit series expression for all coefficients in the virial expansion of the free energy and the experimentally measurable conductance. Some results for correlation functions are also presented. The second method gives an expression for the full free energy, which yields a surprising fluctuation-dissipation relation between the conductance and the free energy.