INFORME DE LA COMISIÓN FISCALIZADORA

The term entropy was introduced by Rudolf Clausius in 1865 to describe certain observations in connection with heat transport and conversion, see Clausius (1865). His descriptions adopted some earlier discoveries, among others by Nicolas Leonard Sadi Carnot and Antoine Laurent de Lavosier, regarding the nature of heat and following debates around the portability of energy. The driving insight was that processes of energy transformation seem to have a clear preferential direction in time. This was in particular a constraint to the enthusiastic perceptions of the energy principle, the first law of thermodynamics, that promised the prospect of unlimited energy sources and convertibility. In other words, while the convertibility of energy contained a promise, regarding the transfer of human labor into material wealth, the ideas of fatigue or entropy rather implied a drawback in terms of potential progress.2

The discussions around the term entropy followed a speculative reasoning about a substance analogously to heat, which itself could be governed by similar balance laws. Thus, rather than resulting solely from precise and quantitative physical measurements, the term entropy also denotes qualitative observations and insights, and determined the establishment of a second law of thermodynamics. This second law or entropy principle basically states that not every form of energy is convertible equally in other forms of energy, so while, for example, mechanical work can be transformed into heat completely, this is not possible in reverse. While there were first statistical formulations for these circumstances by Ludwig Boltzmann and Josiah Willard Gibbs, a formulation in the framework of continuum mechanics was established by Pierre Maurice Marie Duhem with regard to the work and less general formulations of Clausius, thus named the Clausius-Duhem inequality. In this inequality, the entropy is in particular linked to the development of the internal energy (Gurtin & Williams 1966).

2_{Authors like Rabinbach (1992) argue that the formulation of the concept of entropy reflects the} historical upheavals of early modernity, drawing a parallel from the articulation of critique (regarding maldevelopment and contradictions of the new system of factories, social shifts and rapidly growing slums) to a, maybe subliminal, awareness in science and scientific findings for the limitation both of newly developed technical possibilities and natural resources.

The concept of entropy 19 In the theoretical discussions that accompanied the development of thermodynam- ics, in the years following Clausius and Duhem, a distinction was drawn between thermodynamics and thermostatics, particularly with recourse to the work of Gibbs. Differing from thermostatics and their treatment of equilibrium states (or those close to equilibrium), Carl Eckart and Josef Meixner established a theory for the thermody- namics of irreversible processes in the 1940s (Eckart 1940a, Eckart 1940b, Eckart 1940c, Meixner 1943, Eckart 1948). Taking up their work, Clifford Truesdell (Truesdell 1957, Truesdell 1962), extended these thoughts and their implications, establishing a school of thought called rational thermodynamics, especially interested in the mathematical consequences, as well as in the formulation of a continuum mechanical mixture theory. The mixture theory suggested by Truesdell was able to present a systematic approach for multicomponent fluid systems, incorporating previous ad hoc approaches, e.g. of Fick and Jaumann, see Hutter & Wang (2003) and Wilma ´nski (2005).

In the framework of rational thermodynamics, a range of works was concerned with the further investigation of mixtures, see Noll (1958) and Truesdell (1962). Bernhard Coleman and Walter Noll extended the consequences of the second law for continuum mechanics and Truesdell’s mixture theory, so their work pioneered the attempts of thermodynamically consistent derivation, employing the Clausius-Duhem inequality (Coleman & Noll 1963). Many works in the field of mixtures of fluids and plasticity followed their approach, which became known as the Coleman-Noll entropy principle (Coleman & Gurtin 1967, Nunziato & Walsh 1980, Passman, Nunziato & Walsh 1984), also including research on internal state variables and chemical reactions. Of further interest here is in particular the investigation of how thermodynamic restrictions can be applied to a theory of plasticity, see Green & Naghdi (1965), and the prominent work of Goodman & Cowin (1972), establishing a continuum theory for granular media with recourse to the entropy principle. It is also here, in Goodman & Cowin (1972), that the volume fraction is introduced as an additional field including an own balance equation; a concept that was soon picked up in different works.

This approach utilizing the entropy principle, i.e. the entropy inequality, to system- atically derive constraints on constitutive functions was further enhanced by Ingo M ¨uller, switching from the Clausius-Duhem inequality to a more general formulation of the second law, see M ¨uller (1968) as well as M ¨uller (1971a), M ¨uller (1971b). The new approach was then formalized by I-Shih Liu (Liu 1972a), arriving at the algorithmic form which is called the M ¨uller-Liu entropy principle. An overview can be found in M ¨uller & Liu (1984), and, for a comparison between the approaches of Coleman-Noll and M ¨uller-Liu, see Wang & Hutter (1999a) and Triani, Papenfuss, Cimmelli & Muschik (2008).

Since it is generally accepted that the entropy principle must hold, the entropy principle plays a special role in the field of physical modeling. Rather than a constraint on processes, this can be seen as a restriction of the material models, so one can derive constraints in terms of the material behavior; what this means will be unfurled in the next chapter. For more details on the topic of entropy and entropy principles, Hutter & Wang (2003) and Wang & Hutter (2018) are recommended, as well as Ingo M ¨uller’s

20 Fundamentals and physics book (M ¨uller 2007), for an extensive overview both on the history of thermodynamics and the entropy principle, as well as on the arousing philosophical debates.