We study the free anyon fluid by extending the usual Bogoliubov approximation to the case of nonlocal "statistical" interactions. We thereby compute the ground-state energy, the spectrum of low-lying excitations, the static structure factor, and the pair distribution function. The ground-state energy depends crucially on the magnetic dimension of the anyons, a scale that is unavoidably present at the quantum level. The linearly dispersing collective mode in the excitation spectrum indicates superfluidity.