Prigogine model

Much use has been, and continues to be, made of simplified model schemes representative of general classes of chemical or themial feedback. The oregonator and Lengyel-Epstein models for the BZ and CDIMA systems have been given earlier. Pre-eminent among the more abstracted caricature models is the bnisselator introduced by Prigogine and Lefever [47] which has the following fomi ... 

This model, developed by Prigogine and collaborators,53 depends on the premise that the pure components obey the theorem of cor-... 

Another theoretical basis of the superheated liquid-film concept lies on the irreversible thermodynamics developed by Prigogine [43]. According to this theory, irreversible chemical processes would be described (Equation 13.17) by extending the equation of De Donder, provided that simultaneous reactions were coupled in a certain thermodynamic model, as follows ... 

The extension of the cell model to multicomponent systems of spherical molecules of similar size, carried out initially by Prigogine and Garikian1 in 1950 and subsequently continued by several authors,2-5 was an important step in the development of the statistical theory of mixtures. Not only could the excess free energy be calculated from this model in terms of molecular interactions, but also all other excess properties such as enthalpy, entropy, and volume could be calculated, a goal which had not been reached before by the theories of regular solutions developed by Hildebrand and Scott8 and Guggenheim.7... 

From the quantitative point of view, the success of the cell model of solutions was more limited. For example, a detailed analysis of the excess functions of seven binary mixtures by Prigogine and Bellemans5 only showed a very rough agreement between theory and experiment. One should of course realize here that besides the use of the cell model itself, several supplementary assumptions had to be made in order to obtain numerical estimates of the excess functions. For example, it was assumed that two molecules of species and fi interact following the 6-12 potential of Lennard-Jones ... 

Concerning point (b), a generalized theory was developed independently by Prigogine and his co-workers10 11 and by Scott.18 The main idea was to combine the concept of average potential involved in the cell model with the theorem of corresponding states for pure compounds, in such a way that ... 

Once the door was opened to these new perspectives, the works multiplied rapidly. In 1968 an important paper by Prigogine and Rene Lefever was published On symmetry-breaking instabilities in dissipative systems (TNC.19). Clearly, not any nolinear mechanism can produce the phenomena described above. In the case of chemical reactions, it can be shown that an autocatalytic step must be present in the reaction scheme in order to produce the necessary instability. Prigogine and Lefever invented a very simple model of reactions which contains all the necessary ingerdients for a detailed study of the bifurcations. This model, later called the Brusselator, provided the basis of many subsequent studies. 

The first attempts (G. Klein and I. Prigogine, 1953, MSN.5,6,7) were very timid and not very conclusive. They were devoted to a chain of harmonic oscillators. In spite of a tendency to homogenization of the phases, there was no intrinsic irreversibility here, because an essential ingredient is lacking in this model the interaction among normal modes. The latter were introduced as a small perturbation in the fourth paper of the series (MSN.8). 

Thus, Prigogine and Petrosky (PP) introduced the model of a Large Poincare system (EPS). As stated above, the latter is, in fact, a large system, to which the operation of Thermodynamic limit is applied. Clearly, there exists no real system satisfying strictly the definition of a EPS This infinite system is an idealization, on which, by the way, all of statistical mechanics is based. One should thus be more specific about the statement The irreversible processes... cannot be interpreted as approximations of the fundamental laws (statement 1). Quite explicitly, the approximations that are avoided in the PP theory are (a) the arbitrary coarse-graining and (b) the restriction to small parameters. 

SOL. 15.1. Prigogine et V. Mathot, Application of the cell model to the statistical thermodynamics of solutions, J. Chem. Phys. 20, 49-57 (1952). 

TNC.70. 1. Prigogine, From dynamical systems to socio-economic models, in International Symposium on Evolutionary Dynamics and Nonlinar Economics, University of Texas at Austin, 1989. 

TFL.9.1. Prigogine, J. L. Deneubourg, A. De Palma, G. Engelen, etM. Sanglier, Rapportfinal surun modele nonline aire de trafic urbain (Final report on a nonlinear model of urban traffic), Secretariat d Etat a la Region Bruxelloise, et Institut International de Physique et de Chimie Solvay. 

MSE.6. 1. Prigogine et G. Garikian, Sur le modele de I etat liquide de Eennard-Jones et Devonshire (On the Lennard-Jones and Devonshire model of the liquid state), J. Chim. Phys., 45, 273-289 (1948). 

MSE.IO. I. Prigogine et L. Saraga, Sur la tension superficielle et le modele ceUulaire de I etat liquide (On the surface tension and the cell model of liquid state), J. Chim. Phys. 49, 399-407 (1952). 

MSN.69. A. P. Grecos and I. Prigogine, Kinetic and ergodic properties of quanmm systems— The Friedrichs Model, Physica 59, 77-96 (1972). 

MSN.181. E. Karpov, I. Prigogine, T. Petrosky, and G. Pronko, Friedrichs Model with virtual transitions Exact solution and indirect spectroscopy, J. Math. Phys., 41, 118-131 (2000). 

MSN.189.1. Prigogine, E. Karpov, T. Petrosky, and G. Ordonez, Time operator in the Friedrichs model, in Proceedings, XXIV Workshop on High Energy Physics and Field Theory, Protvino, State Research Centre of Russia, Instimte of High Energy Physics, 2001, pp. 274—281. 

We shall be precise regarding the way the various assumptions are made to derive our model. Let p(r, t) be the number density of a gas of stars, called later on the gravitational plasma following Prigogine and Seveme [5]. All the stars have the same mass m, and p is normalized in such a way that... 

The next two important steps in this narrative are considered to be the following (i) the description of the bruxellator by Prigogine and Lefever, who, following on from Turing s work, analyzed theoretically the ingredients that should be present in a model of chemical reactions in order to produce spatial self-organization (Prigogine and Lefever, 1968) (ii) the description of the Belousov-Zhabotinsky (B-Z) reaction. 

I would like to comment on the theoretical analysis of two systems described by Professor Hess, in order to relate the phenomena discussed by Professor Prigogine to the nonequilibrium behavior of biochemical systems. The mechanism of instability in glycolysis is relatively simple, as it involves a limited number of variables. An allosteric model for the phosphofrucktokinase reaction (PFK) has been analyzed, based on the activation of the enzyme by a reaction product. There exists a parameter domain in which the stationary state of the system is unstable in these conditions, sustained oscillations of the limit cycle type arise. Theoretical... 

Finally, the basic model could be also constructed ad hoc just to reproduce the kinetic phenomena observed experimentally in time and in space the well-known examples are the Brusselator or Prigogine-Lefever model (see [2]) and the model by Smoes [7]). Practically any basic model is oriented for a simplest and transparent description of a particular kind of the autowave processes. 

Irregular behaviour of concentrations and the correlation functions observed in the chaotic regime differ greatly from those predicted by law of mass action (Section 2.1.1). Following Nicolis and Prigogine [2], the stochastic Lotka-Volterra model discussed in this Section, could be considered as an example of generalized turbulence. 

---