Exact solution to the 1d one component Coulomb gas at fixed energy

The one dimensional one component plasma has applications to one dimensional particle systems with logarithmic interactions such as charges in a single channel wire or vortex filaments in a fluid convection stream. The exact integral of this plasma in the canonical ensemble with a gaussian confining potential has already been computed. In this paper, I compute the exact volume of the phase space of the plasma of N particles at fixed energy without a confining potential using a microcanonical ensemble and show that, as in the two-dimensional case, it has negative temperature states, suggesting that one dimensional turbulence can occur from vortex/electron clustering.