Nonequilibrium thermodynamics formalism

The question of the efficiency of biological transport systems was examined extensively in the 1960s on the basis of linear nonequilibrium thermodynamics. I think it would be appropriate to give a brief account of the treatment here, especially since Professor Prigogine s early work was the source of most of our ideas at the time. The formal approach of... 

In 1965, Joseph E. Mayer (Sidebar 13.5) and co-workers published a paper [M. Baur, J. R. Jordan, P. C. Jordan, and J. E. Mayer. Towards a Theory of Linear Nonequilibrium Statistical Mechanics. Ann. Phys. (NY) 65, 96-163 (1965)] in which the vectorial character of the thermodynamic formalism was suggested from a statistical mechanical origin. Although this paper attracted little attention at the time, its results suggest how thermodynamic geometry might be traced to the statistics of quantum mechanical phase-space distributions. 

One possible general approach to calculate the conductivity in proton conductors is the phenomenological approach applying the formalism of nonequilibrium thermodynamics [13] (Section 5.7.6) to calculate the FIGURE 8.10 Hydrogen hydrogen diffusion coefficient in oxides [36],... 

First of all relying directly on the second law we will try to give the interpretation of the Prigogine theorem. Taking into account that the traditional variables of equilibrium thermodynamics are the parameters of state and, wishing to reveal the formalized relations between both thermodynamics, let us consider two situations sequentially (1) when some parameters of interaction that hinder the attainment of final equilibrium between the open subsystem and other parts of the isolated system that contains this subsystem are set (2) when flows are taken constant for the flow exchange between the open subsystem and the environment. It is obvious that both situations can be reduced to the case of fixing individual forces which is normally considered in the nonequilibrium thermodynamics. 

The capabilities of MEIS and the models of kinetics and nonequilibrium thermodynamics were compared based on the theoretical analysis and concrete examples. The main MEIS advantage was shown to consist in simplicity of initial assumptions on the equilibrium of modeled processes, their possible description by using the autonomous differential equations and the monotonicity of characteristic thermodynamic functions. Simplicity of the assumptions and universality of the applied principles of equilibrium and extremality lead to the lack of need in special formalized descriptions that automatically satisfy the Gibbs phase rule, the Prigogine theorem, the Curie principle, and some other factors comparative simplicity of the applied mathematical apparatus (differential equations are replaced by algebraic and transcendent ones) and easiness of initial information preparation possibility of sufficiently complete consideration of specific features of the modeled phenomena. 

However, natural systems consist of flows caused by unbalanced driving forces, and hence the description of such systems requires a larger number of properties in space and time. Such systems are away from the equilibrium state, and are called nonequilibrium systems, they can exchange energy and matter with the environment, and have finite driving forces (Figure 2.1). The formalism of nonequilibrium thermodynamics can describe such systems in a qualitative and quantitative manner by replacing the inequalities of classical thermodynamics with equalities. 

Chemical process rate equations involve the quantity related to concentration fluctuations as a kinetic parameter called chemical relaxation. The stochastic theory of chemical kinetics investigates concentration fluctuations (Malyshev, 2005). For diffusion of polymers, flows through porous media, and the description liquid helium, Fick s and Fourier s laws are generally not applicable, since these laws are based on linear flow-force relations. A general formalism with the aim to go beyond the linear flow-force relations is the extended nonequilibrium thermodynamics. Polymer solutions are highly relevant systems for analyses beyond the local equilibrium theory. 

Extended nonequilibrium thermodynamics theory is often applied to flowing polymer solutions. This theory includes relevant fluxes and additional independent variables in describing the flowing polymer solutions. Other contemporary thermodynamic approaches for this problem are GENERIC formalism, matrix method, and internal variables (Jou and Casas-Vazquez, 2001), which are summarized in the following sections. 

We now briefly consider another important aspect of nonequilibrium thermodynamics, namely phase transformations and how they are modelled. Galenko and Jou198 develop a thermodynamic formalism for rapid phase transformations within a diffuse interface of a binary system in which the system is in a state of local nonequilibrium. The phase-field method, in which the phase- field variable O varies smoothly and continuously between one pure phase (in which O = +1) and another (in which -1), is used to derive... 

Such a frame is provided by nonequilibrium thermodynamics. Using the frictional formalism [6] we may express the basic phenomenological equations for an n-component system in the form... 

In this chapter the formalism of nonequilibrium thermodynamics, is reviewed. This formalism is then applied to the theory of isothermal diffusion and electrophoresis. It is shown that this theory is important in determining the relations between the transport coefficients measured by light scattering and those measured by classical macroscopic techniques. Since much of this material is covered in other chapters, this chapter is very brief. Our presentation closely follows that of Katchalsky and Curran (1965). Other books that can be consulted are those of DeGroot and Mazur (1962) and Prigogine (1955). 

It is important to recognize that these equations have been derived for an isother-mal-isobaric system. In the event that there are temperature or pressure gradients, the equations are more complicated. In this case there is a coupling between the concentrations and other hydrodynamic modes. We have investigated this coupling for binary solutions in Section 10.6. The formalism of nonequilibrium thermodynamics enables this to be done systematically for any number of components. 

MOLECULAR FLUX OF THERMAL ENERGY IN BINARY AND MULTICOMPONENT MIXTURES VIA THE FORMALISM OF NONEQUILIBRIUM THERMODYNAMICS... 

The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

Often extended nonequilibrium thermodynamics with maximum-entropy formalism leads to more general expression for the entropy not limited to second order in fluxes. 

The first question was answered by Network Thermodynamics (e.g. Oster et al., 1973 Schnakenberg, 1977 Peusner, 1985) adopting the bond graph technique. This approach benefits from the formal correspondence between certain interpretations of nonequilibrium thermodynamics and of electrical network theory. Adopting the notions of chemical impedance , chemical capacity and chemical inductance , chemical reactions as well transport processes can be represented by networks obeying KirchhofFs current and voltage laws. 

These two relations, called the Saxen relations, were obtained originally by kinetic considerations for particular systems, but by virtue of the formalism of nonequilibrium thermodynamics we see their general validity. 

Lehninger in the 1975 edition of his well-known text on biochemistry [6] states ..at least two general attributes of open systems have considerable significance in biology. .. the most important implication is this in the formalism of nonequilibrium thermodynamics, the steady state, which is characteristic of all smoothly running machinery, may be considered to be the orderly state of an open system, the state in which the rate of entropy production is at a minimum and in which the system is operating with maximum efficiency under the prevailing conditions . 

The first step is to formulate the necessary kinetic equations, and work out values for the appropriate kinetic coefficients from a combination of experiment, theory, and observations on natural rocks. There are several approaches we might use, but the formalism of nonequilibrium thermodynamics is particularly convenient for dealing with metamorphic processes, because it emphasizes the chemical potentials of components, which are often easier to evaluate thcin species concentrations. This paper reviews some ways in which the approach can be used to model the kinetics of metamorphic processes which can be studied in outcrop or thin-section it will not consider regional processes such as heat flow or fluid convection (eg. Reed, 1970). 

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