Extended thermodynamics - consistent in order of magnitude

It was always known that ordinary thermodynamics requires fairly smooth and slowly varying fields. Extended thermodynamics on the other hand is needed for rapidly changing fields with steep gradients. This notion is made explicit in the present paper by assigning orders of magnitude in steepness to the moments which are the field variables of extended thermodynamics. Once a process is deemed to be steep of O(n), the number of field variables may be read off from a table and the field equations are closed, by omission of all higher order terms. The procedure is demonstrated for stationary one-dimensional heat conduction and for heat conduction and one-dimensional motion. An instructive synthetical case of a "one-dimensional gas" is also treated and it is shown in this case how the hyperbolic equations of extended thermodynamics may be regularized - or parabolized - in a rational manner. The theory of O(n) is fully compatible with the entropy principle of that order, but no entropy postulate is needed here, at least not for closure. The theory can be shown to be compatible with an exponential phase density. [PUBLICATION ABSTRACT