Are power-law distributions an equilibrium distribution or a stationary nonequilibrium distribution?

We examine whether the principle of detailed balance holds for the power-law distributions that are generated from the well-known two-variable Langevin equation and the associated Fokker–Planck equations. With the detailed balance and the generalized fluctuation–dissipation relation, we derive analytically the stationary power-law distribution from the Ito’s, Stratonovich’s and Zwanzig’s Fokker–Planck equations, and conclude that the power-law distributions can either be a stationary nonequilibrium distribution or an equilibrium distribution, which depend on information about the form of the diffusion coefficient function and the existence and uniqueness of an equilibrium state.