Prigogine approximation

Concerning statement 1,1 believe that one should first define what exactly is meant by approximation. In La Fin des Certitudes (p. 29), Prigogine rightly attacks the rather widely present view according to which statistical mechanics requires a (brute) coarse graining (i.e., a grouping of the microscopic states into cells, considered as the basic units of the theory). This process is, indeed, an arbitrary approximation that cannot be accepted as a basis of the fundamental explanation of the very real macroscopic irreversible processes. 

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. 

Professor Prigogine showed us the wide variety of phenomena that may appear in a nonlinear reaction-diffusion system kept far from thermodynamic equilibrium. The role of diffusion in these systems is to connect the concentrations in different parts of space. When the process of diffusion is approximated by Fick s law, this coupling is linear in the concentration of the chemicals. 

We first consider the case of a two-component solution (biopolymer + solvent) over a moderately low range of biopolymer concentrations, i.e., C < 20 % wt/wt. The quantities pm x in the equations for the chemical potentials of solvent and biopolymer may be expressed as a power series in the biopolymer concentration, with some restriction on the required number of terms, depending on the steepness of the series convergence and the desired accuracy of the calculations (Prigogine and Defay, 1954). This approach is based on simplified equations for the chemical potentials of both components as a virial series in biopolymer concentration, as developed by Ogston (1962) at the level of approximation of just pairwise molecular interactions ... 

The approximate methods of renormalization for the investigation of phase transitions in degenerate states94 were presented to the conference by Kadanoff and by Brezin. The nonequilibrium statistical methods were discussed by Prigogine,95 followed by Hohenberg who treated critical dynamics. In the third part, Koschmieder discussed the experimental aspects of hydrodynamic instabilities96 Arecchi, the experimental aspects... 

H. Koppel, W. Domcke, and L. S. Cederbaum, in Advances in Chemical Physics, I. Prigogine and S. A. Rice, Eds., Wiley, New York, 1984, Vol. 57, pp. 59-246. Multimode Molecular Dynamics Beyond the Born-Oppenheimer Approximation. 

Tn recent years, developing interests in surface energetics and adhesion of liquid-like polymers, or polymer liquids, have prompted both theoretical and experimental work on surface tension. Unlike low molecular weight liquids, polymer liquids have not been extensively studied. Bondi and Simkin (1) mentioned surface tension in their study on high molecular weight liquids. Roe (28) applied both the cell theory of polymer liquids and the hole theory of surface tension of simple liquids to develop an approximate theory of surface tensions of polymer liquids. His approach has met some degree of success. Notably, both Bondi s and Roes work are somewhat related to the cell theory introduced by Prigogine and... 

The structural approach will also contribute to the analysis of the thermodynamics of nonequilibrium systems. It is the aim and purpose of thermod5mamics to describe structural features of systems in terms of macroscopic variables. Unfortunately, classical thermodynamics is concerned almost entirely with the equilibrium state it makes only weak statements about nonequilibrium systems. The nonequihbrium thermodynamics of Onsager (f), Prigogine (2), and others introduces additional axioms into classical thermodynamics in an attempt to obtain stronger and more useful statements about nonequilibrium systems. These axioms lead, however, to an expression for the driving force of chemical reactions that does not agree with experience and that is only applicable, as an approximation, to small departures from equilibrium. A way in which this situation may be improved is outlined in Section VII. 

As a good approximation, the vapor pressure of the solvent over a drop containing a nonvolatile solute is given by an expression of the same form as the Kelvin relationship (9.17) (Defay and Prigogine, 1966). The partial molar volume, u, is that of the solvent, and Ps is the vapor pressure of Che solvent over a solution with a planar surface. 

This observation constitutes the basic idea of the local equilibrium model of Prigogine, Nicolis, and Misguich (hereafter referred to as PNM). One considers the case of a spatially nonuniform system and deduces from (3) an integral equation for the pair correlation function that is linear in the gradients. This equation is then approximated in a simple way that enables one to derive explicit expressions for all thermal transport coefficients (viscosities, thermal conductivity), both in simple liquids and in binary mixtures, excluding of course the diffusion coefficient. The latter is a purely kinetic quantity, which cannot be obtained from a local equilibrium hypothesis. 

M. Baer, A Review of Quantum Mechanical Approximate Treatments of Three Body Reactive Systems, Advances in Chemical Physics, Vol. 49, eds., I Prigogine and S. A. Rice (Wiley, N.Y., 1982)191. 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

The last of the approximate theories that we wish to mention is that of Prigogine and collaborators (Prigogine with contributions from A. Bellemans and V. Mathot. 1957). This theory combined ideas from the cell theory of solutions and from perturbation theory, both mentioned above. This approach was qualitatively quite successful especially insofar as it correctly predicted the relative signs of the various excess functions of mixing these were incorrectly predicted by most other approximate theories in a number of cases. 

Here, the term A includes the contribution of interaction entropy (Floiy 1970). Such a contribution can be miderstood from the concept of compressible free volume in the fluids. When two fluids are mixed with each other, part of molecules of one species enters the free volume of another species, and then the total volume is not a simple addition of the two individual components. Yamakawa made an approximate estimation from the expansion theory (Yamakawa 1971). Prigogine attributed this contribution to a combinatorial contribution of molecular geometry and a non-combinatorial contribution of molecular stmctures, and proposed an equation-of-state theory (Prigogine 1957b). Hory, Orwell and Vrij further considered the contribution of free volume, and employed separate parameters to describe the hard-core volume and surface contacts of chain units (Flory et al. 1964 Flory 1965 OrwaU and Rory 1967). This work makes the equation of state fit better to the e erimental results, and derives the so-called Flory-Orwell-Vrij equation of state for pure polymers, as given by... 

In the literature of stochastic reaction kinetics it was often assumed that the stationary distributions of chemical reactions were generally Poissonian (Prigogine, 1978). The statement is really true for systems containing only first-order elementary reactions, even when inflow and outflow are taken into account (i.e. for open compartmental systems see Cans, 1960, p. 692). If the model of open compartmental systems is considered as an approximation of an arbitrary chemical reaction near equilibrium, then in this approximation the statement is true. 

Two forms of the cell model (CM) are then developed harmonic oscillator approximation and square-well approximation. Both forms assnme hexagonal closed packing (HCP) lattice structure for the cell geometry. The model developed by Paul and Di Benedetto [13] assumes that the chain segments interact with a cylindrical symmetric square-well potential. The FOV model discnssed in the earlier section uses a hard-sphere type repulsive potential along with a simple cubic (SC) lattice structure. The square-well cell model by Prigogine was modified by Dee and Walsh [14]. They introduced a numerical factor to decouple the potential from the choice of lattice strncture. A universal constant for several polymers was added and the modified cell model (MCM) was a three-parameter model. The Prigogine cell EOS model can be written as follows. 

Statistical thermodynamic theories provide a powerful tool to bridge between the microscopic chemical structures and the macroscopic properties. Lattice models have been widely used to describe the solution systems (Prigogine 1957). Chang (1939) and Meyer (1939) reported the earliest work related with the lattice model of polymer solution. The lattice model was then successfully established by Flory (1941, 1942) and Huggins (1942) to deal with the solutions of flexible polymers by using a mean-field approximation, and to derive the well-known Flory-Huggins equation. 

Another problem is that of data correlation, or of data generalization by means of well founded theory. At the rough approximation level we are (thanks to the work by Flory, Huggins, Prigogine, Prausnitz, and others) in good shape, and heretofore that has been quite adequate. But we shall see that recent studies of concentrated systems exhibit vapor/llquid equilibria which are not even described qualitatively by existing theory, assuming that the measurements are reliable. 

In most cases, in order to estimate the miscibility of two polymers, the approximations of the theories of regular solutions are used. New statistical theories developed by Prigogine, Patterson, Sanchez, and Flory are also widely used. 

Abstract chemical models exhibiting nonlinear phenomena were proposed more than a decade ago. The Brusselator of PRIGOGINE and LEFEVER [54] has oscillatory (limit cycle) solutions, and the SCHLOGL [55] model exhibits bistability, but these models have only two variables and hence cannot have chaotic solutions. At least 3 variables are required for chaos in a continuous system, simply because phase space trajectories cannot cross for a deterministic system. As mentioned in the Introduction, the possibility of chemical chaos was suggested by RUELLE [1] in 1973. In 1976 ROSSLER [56], inspired by LORENZ s [57] study of chaos in a 3 variable model of convection, constructed an abstract 3 variable chemical reaction model that exhibited chaos. This model used as an autocatalytic step a Michaelis-Menten type kinetics, which is a nonlinear approximation discovered in enzymatic studies. Recently more realistic biochemical models [58,59] have also been found to exhibit low dimensional chaos. 

The importance of the c parameter was first recognized by Prigogine et al. [70]. It is to be taken as a measure of the perturbation of internal rotations of the chain in addition to motions of the chain as a whole, in the dense medium. The Flory approximation [12, 13] is invoked to represent the combinatorial entropy... 

The interfacial dissipative structures are examples of selforganization predicted by the theories of stability of non equilibrium system of Prigogine, Glansdorff and Nicolis [11] [12] extended to capillary systems by Steinchen and Sanfeld [15]. We will now give the synthetic framework of the basic equations for charged and polarized surfaces in a pure electrostatic approximation [ 14l ... 

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