Brussels School

Only in the last decades has the thermodynamics of open systems been treated intensively and successfully. The thermodynamics of irreversible systems was studied initially by Lars Onsager, and in particular by Ilya Progogine and his Brussels school both studied systems at conditions far from equilibrium. Certain systems have the capacity to remain in a dynamic state far from equilibrium by taking up free energy as a result, the entropy of the environment increases (see Sect. 9.1). 

The importance of the work of Prigogine and his Brussels school for a deeper understanding of evolution processes is mainly due to the fact that it is possible... 

GEN.278.1. Prigogine, L apport de TEcole de Thermodynamique et de Mecanique statistique de Bruxelles (The contribution of the Brussels school of thermodynamics and statistical mechanics), article redige dans le cadre d un ouvrage sur les activites de TULB a I approche de fan 2000 (non public). 

Classical dynamics and orthodox quantum mechanics are constructed along the model of integrable systems in the sense of Poincare. Our aim is to construct dynamics for nonintegrable systems. As far as we know, this is a new attempt, which has its roots in the early work of the Brussels School [1-9]. The main result is that we have to replace the unitary transformation f/ by a nonunitary... 

Before addressing the main topic of chemical evolution, I would like to discuss briefly the rather curious story of the Belgian school of thermodynamics, often called the Brussels school. It took shape at the end of the 1920s and during the 1930s. At a time when the great schools of thermodynamics, such as the Californian school founded by Lewis and the British school with Guggenheim, directed their efforts almost exclusively to the study of equilibrium systems, the point of view presented by the Brussels school appeared as quite unorthodox and somewhat controversial. Indeed, the Brussels school tried to approach equilibrium as a special case of nonequilibrium and concentrated its efforts on the presentation of thermodynamics in a form that would be applicable also to nonequilibrium situations. This story is rather curious from the point of view of the history of science, so let me go into a little more detail. 

The introduction of affinity by De Donder marks the birth of the Brussels school the first publication appeared around 1922, but it took some years to make these concepts more precise.4 What was the reaction of the scientific community When we go through the proceedings of the Belgian Royal Academy, we see that De Donder s work indeed aroused much local interest. Verschaffel from Ghent and Mund from Louvain were among the people who became active in this newborn nonequilibrium chemical thermodynamics. However, one has to say that elsewhere De Donder s approach met with skepticism and even with hostility. His introduction of affinity was thought of as merely a different notation. 

The net reaction is A I B -> E I F. This reaction scheme has been developed by the Brussels School of Thermodynamics, and consists of a trimolecular collision and an autocatalytic step. This reaction may take place in a well-stirred medium leading to oscillations, or the diffusions of the components A and B may be considered. In the latter case, the system may produce Turing structures. 

As a student and collaborator of Professor de Bonder, Professor R. Defay has contributed to the work of the Brussels School omthe Thermodynamics of chemical reactions. His own researches are chiefly related to azeotropism, to surface tension, to the extension of the phase rule, to capillary systems and to the study of surfaces not in adsorption equilibrium. 

Another hypothetical mechanism of a chemical reaction is the model called Brusselator, investigated by the Brussels school of Prigogine... 

The extent of reaction is a parameter first introduced by De Donder. It has been used systematicsilly by the Brussels School of Thermodynamics. See I. Progogine and R. Defay (tr. D. H. Everett), Chemical Thermodynamics, Longmans, Green and Co., London, 1954. 

To explain the existence of the LC. ST of such a type, a theory of corresponding states (called the theory of liquid state as well) ha.s received a large d< velopment effort by Prigogine et al. (1953), Prigogine (1957), Flory el al. (196dab), Flory (1965, 1970), Patterson and Delmas (1970),, Siow et al. (1972). This approach is baseil on the theory of r-dimcnsional liquids developed by the Brussels school (Ilia Prigogine el al.)... 

This is the formula which, in its generalized form in terms of forward and reverse affinities of elementary reactions, we have called the Marcelin-De Donder formula, in agreement with the literature from the Brussels school ... 

In 1977, Nicolis and Prigogine summarized the work of the Brussels school in a book entitled Self-Organization in Nonequilihrium Systems. For his contributions to the study of nonequilibrium systems, Ilya Prigogine was awarded the 1977 Nobel prize in chemistry. 

This book is the outcome of decades of work. The senior author was a student of Theophile De Donder (1870-1957), the founder of the Brussels School of Thermodynamics. Contrary to the opinion prevalent at that time, De Donder considered that thermodynamics should not be limited to equilibrium situations. He created an active school in Belgium. But his approach remained isolated. Today the situation has drastically changed. There is a major effort going on in the study of nonequilibrium processes, be it in hydrodynamics, chemistry, optics or biology. The need for an extension of thermodynamics is now universally accepted. 

Prigogine s teacher De Bonder was the founder of The Brussels School of Thermodynamics that focused on the thermodynamic treatment of irreversible processes, including chemical reactions. Much of the work of The Brussels School was connected with rates of entropy-production of systems that are not at equilibrium. One of Prigogine s early achievements, after he was appointed to the ULB physical chemistry faculty, was to demonstrate that non-equilibrium systems that are close to equilibrium necessarily will evolve so as to approach the equilibrium state in such a way that the rate of entropy production is as low as is possible. 

The local equilibrium assumption was the basis on which the Brussels school developed a global thermodynamic theory. Use of this assumption makes possible the macroscopic evaluation of entropy production and entropy flow terms with macroscopic thermodynamic methods. The assumption states that "there exists within each small mass element of the medium a state of local equilibrium for which the local entropy, s, is the same function of the local macroscopic variables as at equilibrium state" (Glansdorff and Prigogine, 1971, p. 14). In other words, each small element of a system may be treated as a state near equilibrium but need not necessarily be at equilibrium. This does not mean that the system as a whole need be near equilibrium thus, neighboring local elements may differ in parameters (temperatures, chemical affinities, etc.) which are reflected in the function describing their local entropy. The additional assumption is made that the sum of the criteria of local stability for each element corresponds to the global stability criterion for the whole system. 

Symmetry-Breaking Instabilities The Trimolecular Reaction. The "Brusselator" or trimolecular reaction is the simplest model which exhibits instabilities that may be symmetry-breaking in space and/or time. Although it does not represent an actual chemical reaction, it is nevertheless the best-studied and most widely known theoretical model for chemical instability phenomena. Historically it is the model on which the study of dissipative structures was begun by members of the Brussels. School of Thermodynamics (hence its popular name) a decade ago (44, 45, 46, 47). 

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