A rate equation for a discrete probability distribution is discussed as a route to describe smooth relaxation towards the maximum entropy distribution compatible at all times with one or more linear constraints. The resulting dynamics follows the path of steepest en-tropy ascent compatible with the constraints. The rate equation is consistent with the Onsager theorem of reciprocity and the fluctuation-dissipation theorem. The mathematical formalism was originally developed to obtain a quantum theoretical unification of mechanics and ther-modinamics. It is presented here in a general, non-quantal formulation as a part of an effort to develop tools for the phenomenological treatment of non-equilibrium problems with appli-cations in engineering, biology, sociology, and economics. The rate equation is also extended to include the case of assigned time-dependences of the constraints and the entropy, such as for modeling non-equilibrium energy and entropy exchanges.