1. Heat Conduction

More generally, a quantity is extensive if it has a density with respect to one or another measure of spatial extension of the region a material body occupies, be it the volume, area, or length, measure.

This fact was evident to Gibbs: on p. 7 of [61] we read the following pertinent quotation of Gibbs: “...heat received at one temperature is by no means the equivalent of the same amount of heat received at another temperature ...”.

An interesting discussion of time reversal transformations is found in Sect. 10.3 of [17]; see also [18].

As remarked by Gurtin ([27], p. 53), the Coleman-Noll procedure “...is based on the premise that the second law be satisfied in all conceivable processes, irrespective of the difficulties involved in producing such processes in the laboratory. The rational application of this procedure requires external forces and supplies that may be assigned arbitrarily to ensure satisfaction of the underlying balances in all processes. This may seem artificial, but it is no more artificial than theories based on virtual power, a paradigm that requires arbitrary variations, not guaranteed to be consistent with the resulting evolution equations, granted a constitutive description. The Coleman-Noll procedure makes explicit the external fields needed to support the ‘virtual processes’ used, and in so doing ensures that these external fields, whether virtual or not, enter the theory in a thermodynamically consistent manner.”

Representation (1.35) of the heat influx is granted by an application of a lemma, whose simple proof can be found in [6]:

Lemma. Let V be a finite-dimensional vector space, let \(f_o\in V\), and let f be a \(C^1\)-mapping into V from an open neighborhood \(\mathcal O\) of the origin in V, such thatThen, f has the representationwith(here D denotes differentiation).

Here \(\mathbf {I}\) denotes the identity tensor.

The variational derivative \(\delta F\{P;\vartheta \}\) at \(\vartheta \) of functional F is defined as follows:for all variation fields \(\,\delta \vartheta \;\text {such that}\; \delta \vartheta \equiv 0\;\text {over}\;\partial P\). The stationarity condition on F expressed by the vanishing of its variational derivative for whatever admissible variation leads to the Euler-Lagrange equationIn the case of the Dirichlet functional (1.44), the Euler-Lagrange equation yields, after localization, the archetypal elliptic PDE:associated with the names of Poisson and Laplace.

The decay rate, in addition to the physical constants, depends on the region P in a way that, roughly speaking, can be shown to be smaller and smaller as P gets larger and larger.

For a similar discussion with similar results within the framework of the Green-Naghdi Type III theory of heat propagation, the reader is referred to [5]; see also [4].

The notion of observer change is shortly recapped in the Appendix, Sect. A.4.

For the same reason, in the case of more encompassing theories than pure heat conduction, a control field is needed for each additional balance law.

The corresponding excess inflow is: