Stability of nonequilibrium stationary states

A very general criterion for stability of a state was formulated by Lyapunov [4]. We shall obtain the conditions for the stability of a nonequilibrium state using Lyapunov s theory. 

Lyapunov s formulation gives conditions for stability in precise mathematical terms (with clear intuitive meaning). Let Xg be a stationary state of a physical system. In general, X may be an r-dimensional vector with components Xk, k = 1,2. r. We shall denote the components of Xg by Xg - Let the time evolution of X be described by an equation 

The stability of the stationary state can be understood by looking at the behavior of a small perturbation 6Z. To establish the stability of a state, first a positive function L(8X) of 6X, which may be called a distance, is defined in the space spanned by Xu. If this distance between Xgk and the perturbed state (Xfjt + hXk) steadily decreases in time, the stationary state is stable. Thus state Xsk is stable if 

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. 

Box 18.1 Kinetic Equations and Lyapunov Stability Theory An Example 

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STABILITY OF NONEQUILIBRIUM STATIONARY STATES USING THE STABE.ITY CRITERION... 

Hence (2.17, 2.18) are necessary and sufficient conditions for the existence and stability of nonequilibrium stationary states. 

Since 8S< 0 under both the equilibrium and nonequilibrium conditions, the stability of a stationary state is accomplished if... 

The general theory of thermodynamics of nonequilibrium processes also tells us the stability of nonequiUbrium stationary states with respect to spontaneous fluctuations of the internal thermodynamic parameters in the system. It turns out that this stability can also be investigated by analyzing the variations in the entropy production or energy dissipation rates on drawing the system away from its stationary 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. 

Equations (12.27) and (12.64) show the stability of the nonequilibrium stationary states in light of the fluctuations Sev The linear regime requires P > 0 and dP/dt < 0, which are Lyapunov conditions, as the matrix (dAJdej) is negative definite at near 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. 

These conditions ensure the stability of the nonequilibrium stationary states in... 

In 1977. Professor Ilya Prigogine of the Free University of Brussels. Belgium, was awarded Ihe Nobel Prize in chemistry for his central role in the advances made in irreversible thermodynamics over the last ihrec decades. Prigogine and his associates investigated Ihe properties of systems far from equilibrium where a variety of phenomena exist that are not possible near or al equilibrium. These include chemical systems with multiple stationary states, chemical hysteresis, nucleation processes which give rise to transitions between multiple stationary states, oscillatory systems, the formation of stable and oscillatory macroscopic spatial structures, chemical waves, and Lhe critical behavior of fluctuations. As pointed out by I. Procaccia and J. Ross (Science. 198, 716—717, 1977). the central question concerns Ihe conditions of instability of the thermodynamic branch. The theory of stability of ordinary differential equations is well established. The problem that confronted Prigogine and his collaborators was to develop a thermodynamic theory of stability that spans the whole range of equilibrium and nonequilibrium phenomena. 

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. 

When a reactive system is far from its thermodynamic equilibrium, corol laries of the Prigogine theorem, which were derived for the case of the linear nonequilibrium thermodynamics, cannot be applied to analysis directly. Nevertheless, tools of thermodynamics of nonequilibrium pro cesses allow the deduction of some important conclusions on properties of the system, even though strongly nonequilibrium, including in some cases on the stability of stationary states of complex stepwise processes. For several particular cases, theorems similar to the Prigogine theorem can be proved, too. 

If the kinetic equations of the systems are known, there is a well-defined mathematical method to determine at what point the stationary state will become unstable. This is the linear stability analysis that we will discuss in the following section. Nonequilibrium instabilities give rise to a great variety of structures, which we will discuss in the next chapter... 

Some nonequilibrium phase transitions involve the passage of a system from a stationary (time-independent) nonequilibrium state to a nonstationary nonequilibrium state with rich spatiotemporal order. For example, it has been observed that certain chemically reactive systems can pass from a quiescent homogeneous state to a state characterized by spatial and/or temporal oscillations in the concentrations of certain chemical species. In nonequilibrium thermodynamics, Prigogine has argued that such phase transitions result from the violation of the stability condition... 

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