Stability of equilibria

The conditions for mechanical instability can be derived from a set of criteria for the stability of equilibrium systems put forward by Gibbs [8], Considering instability with regard to temperature and pressure, the criteria for stability are... 

The stability of equilibrium points is determined by standard stability analysis (cf. Appendix A). The Jacobian matrix of the linearized system,... 

The general condition for the stability of equilibrium state with respect to thermal, volume, and fluctuations in the numbers of moles is obtained by combining Eqs. (12.7), (12.9), and (12.12)... 

Table 12.1 Necessary and sufficient conditions for the stability of equilibrium state...

The term to the right of the equal sign in Eq. (12.32) is the excess entropy production. Equations (12.31) and (12.32) describe the stability of equilibrium and nonequilibrium stationary states. The term 82S is a Lyapunov functional for a stationary state. 

It is possible to study stability of equilibrium points for this Hamiltonian [62]. Because of symmetry, we know beforehand that the equilibrium point may exist only at z = 0 and either at x = 0,y = or at the two equivalent points turned 120° in the z = 0 plane. These three equilibrium configurations correspond to the three situations of Fig. 18. 

Applicfttion to ft gas that obeys Mariotte s law, 11.—8. In some cases, the work of the forces applied to a system depends only on the initial and final states of this system, 12.—. In general, the work done by the forces applied to a system depends upon every modification the system undergoes, 12.—10. Potential, 14.—11. Potential due to gravity, 16.—la. Forces which admit a potential in virtue of the restrictions imposed upon the system, 16.—13. Energy, 16.—14. Principle of virtual displacements, 17.—15. Conservatives of energy. Conservative systems, 18.—16. Principle of virtual displacements for conservative systems. Stability of equilibrium, 19. 

Stability of equilibrium, 96.—8a. Interpretation of the non-compensated work, 97.—83. Intensity of reaction slow reactions, 98.-84. Very intense reactions principle of maximum work, 99.—... 

Le Chatelier s principle is a qualitative way of describing the stability of equilibrium states against sudden perturbations in concentration, pressure, and temperature. The responses of the system to all three effects can be described quantitatively by thermodynamics. Here we describe the effect of temperature, which is the most useful of these quantitative descriptions. 

In the previous sections of this chapter we have developed criteria for states of thermodynamic equilibrium in various systems. In this section we shall discuss the stability of equilibrium states and present some conclusions concerning the properties of thermodynamic variables in stable, equilibrium systems. 

Summary. This section shows the analysis of equilibrium state for a given system (single linear fluid in this case), which can be made once its final constitutive equations were derived. The equilibrium is defined so as to give the zero entropy production, cf. (3.220)-(3.222). To ensure the persistence of equiUbrium (see the property S4 in Sect. 1.2), the regularity conditions (3.232)-(3.234) were added to the model of linear fluid. The majority of this section was devoted to the analysis of the stability of equilibrium the concept of stable equilibrium was explained on page 127. The condition of stability called the Gibbs stability are (3.256) and (3.257). We... 

Coleman, B.D. On the stability of equilibrium states of general fluids. Arch. Ration. Mech. Anal. 36(1), 1-32 (1970)... 

Definition of equilibrium here is difficult to achieve in practice because of molecular fluctuations in fact the stability of equilibrium, i.e. its return back after its disturbance, must be achieved and thus the equilibrium may be realized. The problem of the stability of equilibrium will be discussed in the remaining part of this Sect.4.7 proceeding similarly as in Sect. 3.8, although the problem is more complicated mainly due to chemical reactions, cf. [39, 98, 143, 146, 147],... 

Equilibrium discussions are more simple as was noted in Sect.4.7 e.g. regularity (giving usual stability of equilibrium) demands inequality only in (4.403). For the case n = 1 we recover the results for the single linear fluid of Sect. 3.6. 

Chapter 3 adds also the description of spatial distribution (gradients). Only single fluid is considered for the sake of simplicity and preparation of the basics for the subsequent treatment of mixtures. Mathematics necessary for the spatial description is introduced in Sect. 3.1. Section 3.2 in the same chapter stresses the importance of the referential frame (coordinate system) and its change in the mathematical description. Sections 3.3—3.6 shows the development of final material model (of a fluid) within our thermodynamic framework, consistent with general laws (balances) as well as with thermodynamic principles (the First and Second Laws and the principles of rational thermodynamics). The results of this development are simplified in Sect. 3.7 to the model of (single) fluid with linear transport properties. Sections 3.6 and 3.7 also show that the local equilibrium hypothesis is proved for fluid models. The linear fluid model is used in Sect. 3.8 to demonstrate how the stability of equilibrium is analysed in our approach. 

The problems of simultaneously treating spatial distributions of both temperature and concentration are currently the concern of the chemical engineer in his treatment of catalyst particles, catalyst beds, and tubular reactors. These treatments are still concerned with systems that are kineticaliy simple. The need for a unified theory of ignition has been highlighted by contemporary studies of gas-phase oxidations, many features being revealed that neither thermal theory, nor branched-chain theory for that matter, can resolve alone. A successful theoretical basis for such reactions necessarily involves the treatment of both the enorgy balance and mass balance equations. Such equations are invariably coupled and cannot be solved independently of each other. However, much information is offered by the phase-plane analj s of the syst (e.g. stability of equilibrium solutions, existence of oscillations) without the need for a formal solution of the balance equations. 

From the point of view of concepts, the problem of the adherence of elastic solids is completely solved only three ingredients are needed geometry of the system, elastic properties, and energy for adhesion. Eventually a fourth ingredient must be added the rigidity of the measuring apparatus that can change the stability of equilibrium. 

The principle of maximum entropy dS) = 0 and (fS) < 0, and the principle of minimum internal energy dU) = 0 and d U) > 0, are the fundamental principles of stable equilibrium, and can be associated with thermodynamic stability, although the stability of equilibrium is not unique to thermodynamics. 

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. 

Eqs. (103a) and (103b) do not provide information on the stability of equilibrium. Stability of equilibrium is recognized only when taking into account the variations of degrees greater than unity, S S, S S,... or 8 U, S U,... in the total variations. 

Gavalas (1968) was an early pioneer in the treatment of the deterministic models of chemical reaction kinetics. His book deals with homogeneous systems and systems with diffusion as well. Basing himself upon recent results in nonlinear functional analysis he treats such fundamental questions as stoichiometry, existence and uniqueness of solutions and the number and stability of equilibrium states. Up to that time this treatise might be considered the best (although brief and concise) summary of the topic. 

Palamodov, V. P. On stability of equilibrium in a potential field. Funkts. Analiz i yego Prilozhen., II, issue 4 (1977), 42-55. 

As we have emphasized in the preceding section, the stability of equilibrium states crucially depends on the validity of dL/dt 0 which is a direct consequence of the second law of thermodynamics only within the range of the linear relationships between the fluxes I and the forces F. Since in general this linearity will not be valid in the vicinity of steady states arbitrarily far from equilibrium, we cannot transfer the above stability proof to such states. 

---