NEW APPROACH TO CONSTRAINED-MAXIMIZATION NONEQUILIBRIUM PROBLEMS.

Many nonequilibrium problems in thermodynamics, information theory, economics, control theory and other sciences require the description of a smooth constrained evolution towards an equilibrium state maximizing some function of the state variables. We present a novel mathematical formalism designed to model a wide class of such nonequilibrium problems. It is based on a type of nonlinear evolution equation with the following main relevant properties. Its solutions maintain invariant the value of each of the constraints like, for a typical thermodynamic nonequilibrium problem, the energy and the number of particles of each species. It maintains a positive-definite time rate of change of the function of the state variables to be maximized like, in the thermodynamic example, the entropy function. Its solutions converge towards an equilibrium state maximizing that function subject to the values of the constraints. A byproduct of the mathematical formalism is a numerical scheme for nonlinear optimization under nonlinear constraints similar to the well-known gradient-projection and steepest-ascent methods of nonlinear programming. However, the principal application of the formalism is to provide a tool for the time-dependent description of nonequilibrium states and their approach to equilibrium.