The Constructive Role of Irreversible Processes

In nonequilibrium systems, oscillating concentrations and geometrical concentration patterns can be a result of chemical reactions and diffusion, the same dissipative processes that, in a closed system, wipe out inhomogeneities and drive the system to a stationary, timeless homogeneous state of equilibrium. Since the creation and maintenance of organized nonequilibrium states are due to dissipative processes, they are called dissipative structures [3]. 

The two concepts of dissipative structures and order through fluctuations encapsulate the main aspects of nonequilibrium order that we describe in this chapter. 

In the previous chapter we have seen that the stability of the thermodynamic branch is no longer assured when a system is driven far from equilibrium. In section 18.3 we have seen how a necessary condition (18.3.7) for a system to become unstable can be obtained by using the second variation of entropy, 5 5. Beyond this point, we are confronted with a multiplicity of states and unpredictability. To understand the precise conditions for instability and the subsequent behavior of a system, we need to use the specific features of the system, such as the rates of chemical reactions and the hydrodynamic equations. There are, however, some general features of far-from-equilibrium systems that we will summarize in this section. A detailed discussion of dissipative structures will be presented in the following sections. 

The loss of stability of a nonequilibrium state can be analyzed using the general theory of stability for solutions of a nonlinear differential equation. Here we encounter the basic relationship between the loss of stability, multiplicity of solutions and symmetry. We also encounter the phenomenon of bifurcation or branching of new solutions of a differential equation from a particular solution. We shall first illustrate these general features for a simple nonlinear differential equation and then show how they are used to describe far-from-equilibrium systems. 

AN ELEMENTARY EXAMPLE OF BIFURCATION AND SYMMETRY BREAKING Consider the equation 

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