Extremum Principles for Irreversible Processes

Abstract

A general problem is considered for the determination of the generalized fluxes and the conjugate generalized forces describing irreversible processes in nonequilibrium thermodynamics. It is shown that for linear systems the solution of this problem is also the solution of a minimum problem and of a maximum problem. In certain cases the functional which is minimized is the rate of entropy production, while in other cases the functional which is maximized is minus this rate. Thus, in these cases Prigogine's principle of the minimum rate of entropy production is valid. For certain dynamical systems it is also shown that the functional in the minimum problem is a decreasing function of time, while for other systems the functional in the maximum problem is an increasing function of time. The results are applied to a system of chemical components in which various chemical reactions, diffusion, and heat conduction are occurring. Similar results are obtained for special nonlinear systems of this kind and for certain linear systems in which convection is also occurring. The results are based, in part, upon the theory of reciprocal variational problems, which is shown to follow from Fenchel's duality theorem involving convex functions.