Nonequilibrium systems stability

The two main assumptions underlying the derivation of Eq. (5) are (1) thermodynamic equilibrium and (2) conditions of constant temperature and pressure. These assumptions, especially assumption number 1, however, are often violated in food systems. Most foods are nonequilibrium systems. The complex nature of food systems (i.e., multicomponent and multiphase) lends itself readily to conditions of nonequilibrium. Many food systems, such as baked products, are not in equilibrium because they experience various physical, chemical, and microbiological changes over time. Other food products, such as butter (a water-in-oil emulsion) and mayonnaise (an oil-in-water emulsion), are produced as nonequilibrium systems, stabilized by the use of emulsifying agents. Some food products violate the assumption of equilibrium because they exhibit hysteresis (the final c/w value is dependent on the path taken, e.g., desorption or adsorption) or delayed crystallization (i.e., lactose crystallization in ice cream and powdered milk). In the case of hysteresis, the final c/w value should be independent of the path taken and should only be dependent on temperature, pressure, and composition (i.e.,... 

In the preceding sections, various types of fluctuations and instabilities essential to corrosion were examined. As a result, it was shown that a corrosion system involves various kinds of problems of stability and instability. Unlike thermodynamic equilibrium systems, in nonequilibrium systems like corrosion systems, a drastic change in the reaction state should be defined as a bifurcation phenomenon. 

In contrast to a mixture of redox couples that rapidly reach thermodynamic equilibrium because of fast reaction kinetics, e.g., a mixture of Fe2+/Fe3+ and Ce3+/ Ce4+, due to the slow kinetics of the electroless reaction, the two (sometimes more) couples in a standard electroless solution are not in equilibrium. Nonequilibrium systems of the latter kind were known in the past as polyelectrode systems [18, 19]. Electroless solutions are by their nature thermodyamically prone to reaction between the metal ions and reductant, which is facilitated by a heterogeneous catalyst. In properly formulated electroless solutions, metal ions are complexed, a buffer maintains solution pH, and solution stabilizers, which are normally catalytic poisons, are often employed. The latter adsorb on extraneous catalytically active sites, whether particles in solution, or sites on mechanical components of the deposition system/ container, to inhibit deposition reactions. With proper maintenance, electroless solutions may operate for periods of months at elevated temperatures, and exhibit minimal extraneous metal deposition. 

We review in Section II the basic results of nonequilibrium thermodynamic stability theory and recall the thermodynamic and kinetic conditions necessary to the occurrence of cooperative coherent behaviors in chemical systems. We briefly indicate some known experimental systems that meet these requirements and in which dissipative structures... 

Beyond the domain of validity of the minimum entropy production theorem (i.e., far from equilibrium), a new type of order may arise. The stability of the thermodynamic branch is no longer automatically ensured by the relations (8). Nevertheless it can be shown that even then, with fixed boundary conditions, nonequilibrium systems always obey to the inequality1... 

Order arising through nucleation occurs both in equilibrium and nonequilibrium systems. In such a process the order that appears is not always the most stable one there are often competing processes that will lead to different structures, and the structure that appears is the one that nucleates first. For instance, in the analysis of the different possible structures in diffusion-reaction systems17-20 one can show, by analyzing the bifurcation equations, that there are several possible structures and some of them require a finite amplitude to become stable if this finite amplitude is realized through fluctuation, this structure will appear. In the formation of crystals (hydrates) the situation is similar the structure that is formed depends, according to the Ostwald rule, on the kinetics of nucleation and not on the relative stability. 

The Gibbs stability theory condition may be restrictive for nonequilibrium systems. For example, the differential form of Fourier s law together with the boundary conditions describe the evolution of heat conduction, and the stability theory at equilibrium refers to the asymptotic state reached after a sufficiently long time however, there exists no thermodynamic potential with a minimum at steady state. Therefore, a stability theory based on the entropy production is more general. 

Strictly speaking, soils are always nonequilibrium systems. With care, however, a partial equilibrium or steady state can be attained by assuming that the soil solids do not change. This is the usual assumption in cation exchange and adsorption studies. Kittrick and co-workers were able to obtain near-equilibrium measurements of some soil minerals in studies requiring -several years. From the resulting ion activities in solution, they were able to calculate some of the equilibrium constants used for the mineral stability diagrams shown later in this book. 

Most of the traditional adsorption studies of surfactants correspond to dilute systems without aggregation in the bulk phase. At the same time micellar solutions are much more important from a practical point of view. To estimate the equilibrium adsorption, neglecting the effect of micelles can usually lead to reasonable results. However, the situation changes for nonequilibrium systems when the adsorption rate can increase by orders of magnitude when the of surfactant concentration is increased beyond the CMC. Current interest in the adsorption from micellar solutions is mainly caused by recent observations that the stability of foams and emulsions depends strongly on the concentration in the micellar region [1]. This effect can be explained by the influence of the micellisation rate on the adsorption kinetics. 

Thermodynamics plays an important role in the stability analysis of transport and rate processes, and the nonequilibrium thermodynamics approach in particular may enhance and broaden this role. This chapter reviews stability analysis based on the conventional Gibbs approach and tbe nonequilibrium thermodynamics theory. It considers the stability of equilibrium, near-equilibrium, and far-from-equilibrium states with some case studies. The entropy production approach for nonequilibrium systems appears to be more general for stability analysis. One major implication of the nonequilibrium thermodynamics theory is the introduction of distance from global equilibrium as a constraint for determining the stability of nonequilibrium systems. When a system is far from global equilibrium, the possibility of new organized structures of matter arise beyond an instability point. 

The random motion of molecules causes all thermodynamic quantities such as temperature, concentration and partial molar volume to fluctuate. In addition, due to its interaction with the exterior, the state of a system is subject to constant perturbations. The state of equilibrium must remain stable in the face of all fluctuations and perturbations. In this chapter we shall develop a theory of stability for isolated systems in which the total energy U, volume V and mole numbers Nk are constant. The stability of the equilibrium state leads us to conclude that certain physical quantities, such as heat capacities, have a definite sign. This will be an introduction to the theory of stability as was developed by Gibbs. Chapter 13 contains some elementary applications of this stability theory. In Chapter 14, we shall present a more general theory of stability and fluctuations based on the entropy production associated with a fluctuation. The more general theory is applicable to a wide range of systems, including nonequilibrium systems. 

A function L that satisfies (18.3.3) is called a Lyapunov function. If the variables are functions of position (as concentrations in a nonequilibrium system can be), L is called a Lyapunov functional—a functional is a mapping of a set of functions to a number, real or complex. The notion of stability is not restricted to stationary states it can also be extended to periodic states [4]. However, since we are interested in the stability of nonequilibrium stationary states, we shall not deal with the stability of periodic states at this point. 

When a nonlinear system ewolwes under far-from-equilibrium conditions in the vicinity of a bifurcation point, a purely deterministic description often proved to be incomplete. The fluctuations of the dynamical variables can play an essential role and obstruct the observation of a transition expected by a deterministic analysis. In the framework of the deterministic approach, the stability of the different states according to the values of the control parameters is studied through a mathematical analysis of the velocity field. In particular, the theory of normal forms leads to the determination of the various kinds of attractors [l,2]. As far as we are concerned with the stochastic approach, the rrLa te.n. equation, has been widely used to analyze bifurcations of homogeneous or spatially ordered steady states or of limit cycles [3,4]. Our aim in the present contribution is to insist on the generality of the method to analyze various kinds of bifurcations in nonlinear nonequilibrium systems. The general procedure proposed to obtain a local description of the probability, which allows us to determine the system s attractors, turns out to display marked analogies with the theory of normal forms. 

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