The thermodynamics of irreversible processes is normally limited to processes that can be adequately described by linear constitutive relations, like those of Fourier and Newton in a simple gas. In this paper we use thermodynamic arguments to derive the (nonlinear) Burnett equations for a monatomic gas, thus avoiding the complicated kinetic theory by which the equations were discovered and which somewhat obscures the origin of the various terms in the equations. Expressions are given for the entropy, its flux and its production rate correct to second-order in Knudsen number. The theory involves five phenomenological parameters, and as there are eleven coefficients in the second-order terms of Burnett's equations, we are able to deduce several necessary constraints between these coefficients. Compact forms for the equations are found that clarify their physical significance. The general method we have developed is applicable to media other than simple gases.

In a final section we use our theory of Burnett's equations to draw some general conclusions concerning the second law of thermodynamics. It is shown that the Clausius-Duhem inequality holds only for the linear theory of constitutive relations; and that axiomatic generalizations of the inequality to nonlinear processes – common in continuum mechanics–fail because the vital distinction between reversible and irreversible processes is not made.