How to find the (strongly non-Boltzmann) distribution in a far-from-equilibrium system is a problem of long standing. It appears in many different contexts, with topical examples including stochastic resonance and Brownian ratchets. One of the most promising approaches to the problem is through asymptotic analysis of the Fokker-Planck equation in the limit of small noise intensity. In simulations and experiments on real systems, however, the noise intensity is necessarily finite. Corrections to allow for finite noise intensity have recently been introduced for the particular case of escape. We are currently investigating the non-equilibrium distribution over the whole of phase space, for two model systems: the periodically driven, overdamped, Duffing oscillator and the inverted van der Pol oscillator. A modified Monte Carlo technique is being applied to investigate the limit of very small noise intensities. The next-to-leading order of approximation of the solution of the Fokker-Planck equation is used to compare the numerical results with the theory. We show, in particular, how changes in the non-equilibrium probability distribution induced by finite noise intensity are linked to an observable modification in the pattern of optimal paths. The numerical observations are in good agreement with theory.