Prigogine’s theory

It was therefore useful and interesting to compare explicitly the derivations of Bogolubov, Choh-Uhlenbeck, and Cohen (which we shall call the streaming operators method) and the results of Prigogine s theory. Part of this comparison has been made previously. 

However, some problems remain unsolved. The three- and four-body results of Choh and Uhlenbeck and of Cohen, respectively, have to be compared with the corresponding expressions in the Prigogine s theory. Furthermore, for any concentration, one has to see how the systematic generalization of the Boltzmann equation derived by Cohen is related to the long-time evolution equation in Prigogine s theory. The aim of this work is to throw some light on these points. 

Section III is devoted to Prigogine s theory.14 We write down the general non-Markovian master equation. This expression is non-instantaneous because it takes account of the variation of the velocity distribution function during one collision process. Such a description does not exist in the theories of Bogolubov,8 Choh and Uhlenbeck,6 and Cohen.8 We then present two special forms of this general master equation. On the one hand, when one is far from the initial instant the Variation of the distribution functions becomes slower and slower and, in the long-time limit, the non-Markovian master equation reduces to the Markovian generalized Boltzmann equation. On the other hand, the transport coefficients are always calculated in situations which are... 

In Section IV, we develop the former results and we study the structure of the transport operator and of the generalized Boltzmann operator. We also analyse the irreducibility condition which appears in Prigogine s theory by using the graphs of equilibrium statistical mechanics. 

V. EQUIVALENCE BETWEEN THE STREAMING OPERATORS METHOD AND PRIGOGINE S THEORY... 

We shall demonstrate explicitly the equivalence between the results of Bogolubov, Choh and Uhlenbeck, and Cohen (BCUC) and the generalized Boltzmann equation in Prigogine s theory. But it seems useful to us to indicate beforehand some qualitative arguments which allow a physical understanding of the grounds on which this equivalence rests (for more details see ref. 24). 

In the dilute gas case, we can easily establish the equivalence between the results of BCUC and Prigogine s theory. 

C.H. Obcemea, E. Brandas, Analysis of Prigogine s Theory of Subdynamics, Ann. Phys. 

Stationary state—Let us consider the stationary state of the first order when AT is kept constant. Using Prigogine s theory of minimum entropy production [5], we have the following three conditions for the stationary state... 

Erdi, P. Toth, J. (1979). Some comments on Prigogine s theories. React. Kinet. 

Prigogine s theory of dissipative structures generated by a combination of nonlinear chemical reactions and diffusion processes (Glansdorff and Prigogine, 1971 Nicolis and... 

Donahue and Prausnitz [23] developed the perturbed-hard chain theory (PHCT) based on perturbed hard-sphere theory for small molecules and Prigogine s theory for chain molecules. In order to account for attractive and repulsive forces among molecules, empirical parameters such as the c parameter were introduced by subsequent investigators. More accurate expressions for the repulsive and attractive forces were... 

Edens, 199l[ B. Edens Semigroups and Symmetry An Investigation of Prigogine s Theories, Thesis KUL, Leuven,1991. http //philsci-archive.pitt.edu/eirchive/00000436/... 

Flory (11) improved the notation and form of Prigogine s expressions, and it is essentially the Flory form of Prigogine s free-volume theory that is of most use for design purposes. The Flory work (11) leads to an equation of state which obeys the corresponding-states principle ... 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

If this expression is substituted in Eqs. (25) and (92), we obtain exactly the n — 2 term of the generalized Boltzmann equation (see Eqs. 85 and 88) as it appears in the Prigogine formalism. This result is thus equivalent to the formulae (24) and (25) of Cohen s theory from which we started. 

Let us put this result in the generalized Boltzmann equation (92) as derived by Cohen. Remembering (A.44), we get identically the expression (85) which is the generalized Boltzmann equation in Prigogine s formalism. This completes the proof of the equivalence between the two theories. 

This principle is very general, relating neither to the linearity nor to the symmetry of the transport laws. On the other hand, it is difficult to attribute a physical meaning to dxP- The authors later attempted to derive a local potential from this property, and they applied this concept to the study of the chemical and hydrodynamical stability (e.g., the Benard convection). The results of this approach were published in Glansdorff and Prigogine s book Thermodynamic Theory of Structure, Stability and Fluctuations (LS.IO, 10a), published in 1971. 

A first purpose consisted only of generalizing the domain of validity of the theory developed during the years 1956-1970. Prigogine s ambition was to... 

MSN.52.1. Prigogine, Quantum theory of Dissipative Systems, Nobel Symposium 5, S. Claesson, ed.. Interscience, New York, 1967, pp. 99-129. 

MSN.64. I. Prigogine, G. Nicolis, and P. Allen, Eyring s theory of viscosity of dense media and nonequilibrium statistical mechanics, in Chemical Dynamics, Papers in Honor of H. Eyring, Hirshfelder, ed., Wiley, New York, 1971. 

In the linear nonequilibrium thermodynamics theory, the stability of stationary states is associated with Prigogine s principle of minimum entropy production. Prigogine s principle is restricted to stationary states close to global thermodynamic equilibrium where the entropy production serves as a Lyapunov function. The principle is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. 

The stability of transport and rate systems is studied either by nonequilibrium thermodynamics or by conventional rate theory. In the latter, the analysis is based on Poincare s variational equations and Lyapunov functions. We may investigate the stability of a steady state by analyzing the response of a reaction system to small disturbances around the stationary state variables. The disturbed quantities are replaced by linear combinations of their undisturbed stationary values. In nonequilibrium thermodynamics theory, the stability of stationary states is associated with Progogine s principle of minimum entropy production. Stable states are characterized by the lowest value of the entropy production in irreversible processes. The applicability of Prigogine s principle of minimum entropy production is restricted to stationary states close to global thermodynamic equilibrium. It is not applicable to the stability of continuous reaction systems involving stable and unstable steady states far from global equilibrium. The steady-state deviation of entropy production serves as a Lyapunov function. 

Systems-level approaches were defined more than 60 years ago with the application of general systems theory to various scientific fields including chemistry and life sciences with emphasis on morphogenesis (von Bertalanffy 1968 Friboulet and Thomas 2005). In the sixties, concepts derived from Ilya Prigogine s dissipative structures theory were also applied to biochemical oscillation and morphogenesis (Friboulet and Thomas 2005). In the seventies, numerical analyses of biochemical systems were developed which could represent a first step into modem systems biology (Hammer et al. 2004 Aderem 2005). 

During the 1950s and the 1960s, two important theories of the liquid state were developed, initially for simple liquids and later applied to mixtures. These are the scaled-particle theory, and integral equation methods for the pair correlation function. These theories were described in many reviews and books. In this book, we shall only briefly discuss these theories in a few appendices. Except for these two theoretical approaches there has been no new molecular theory that was specifically designed and developed for mixtures and solutions. This leads to the natural question why a need for a new book with the same title as Prigogine s ... 

It is along this route that important developments have been achieved specifically for solutions, providing the proper justification for a new book with the same title. Perhaps a more precise title would be the Local Theory of Solutions. However, since the tools used in this theory are identical to the tools used in Prigogine s book, we find it fitting to use the same title for the present book. Thus, the tools are basically unchanged only the manner in which they are applied were changed. 

Their derivation consisted in expanding the pressure in a power series of the parameter A, and was valid only for small values of A. The procedure which will now be adopted in this paper is not so refined as de Boer and Blaisse s theory, but it will not be required that A is small. The procedure is the same as that adopted by Prigogine and Philippot27 in their theory of energies of liquid helium, and it will be applied to other liquids than liquid helium. These substances are, of course, solid at 0° K except for helium. However, it will be assumed that the same method can be applied to these substances in the following treatment, considering the comparative crudeness of our theory which does not discriminate between liquid and solid states. 

In Prigogine and Philippot s theory which was applied to liquid helium, the structure of the liquid is considered to be of a latticelike form, and the energy at 0° K is assumed to be the sum of the potential energy lattice points of a face-centered cubic lattice, and the zero-point energy Et for the motion of molecules. It is very easy to show that 0 is given by... 

M. Merchan, in I. Prigogine, S.A. Rice (Eds.), Multiconfigurational perturbation theory Applications in electronic spectroscopy. Advances in chemical physics New methods in computational quantum mechanics, Vol. XCIII, Wiley, New York, 1996, pp. 219-331. 

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