Apparent exponential surface density profiles are nearly universal in galaxy discs across Hubble types, over a wide mass range, and a diversity of gravitational potential forms. Several processes have been found to produce exponential profiles, including the actions of bars and spirals, and clump scattering, with star scattering a common theme in these. Based on reasonable physical constraints, such as minimal entropy gradients, we propose steady-state distribution functions for disc stars, applicable over a range of gravitational potentials. The resulting surface density profiles are generally a power-law term times a Sérsic-type exponential. Over a modest range of Sérsic index values, these profiles are often indistinguishable from Type I exponentials, except at the innermost radii. However, in certain parameter ranges, these steady states can appear as broken, Type II or III profiles. The corresponding velocity dispersion profiles are low-order power laws. A chemical potential associated with scattering can help understand the effects of long-range scattering. The steady profiles are found to persist through constant velocity expansions or contractions in evolving discs. The proposed distributions and profiles are simple and solve the stellar hydrodynamic equations. They may be especially relevant to thick discs that have settled to a steady form via scattering.