Structurally stable equilibrium state

Obviously, any statement concerning a real system derived from an analysis of its theoretical idealization, i.e. from its model, must not be too sensible to small uncontrolled variations of the parameters. Hence, it is a standard requirement that one must consider not only a stand-alone system but must also understand what happens with all neighboring systems. This works well for rough (structurally stable) equilibrium states and periodic orbits in this case the qualitative structure is not modified by small perturbations of the right-hand side of the system. In contrast, for systems on the stability boundary, the analysis of close systems may become a real problem. 

C.3. 21. Prove that if the origin is a structurally stable equilibrium state of the system (C.3.3), then the corresponding fixed point of the map (C.3.2) is structurally stable as well. Furthermore, show that the topological types of the equilibrium state of (C.3.3) and the fixed point of (C.3.2) are the same. ... 

Theorem C.2. Structurally stable equilibrium states of the averaged system correspond to structurally stable periodic orbits of the original system ifx is a structurally stable equilibrium state in (C.3.9), then the Poincare map (C.3.8) for the system (C.3.7) has a structurally stable fixed point close to x for all sufficiently small p. ... 

Theorem C.3. (Averaging Theorem) If a is an integer, then for sufficiently small fi> 0 structurally stable equilibrium states of the system... 

Belouzov-Zhabotinsky reaction [12, 13] This chemical reaction is a classical example of non-equilibrium thermodynamics, forming a nonlinear chemical oscillator [14]. Redox-active metal ions with more than one stable oxidation state (e.g., cerium, ruthenium) are reduced by an organic acid (e.g., malonic acid) and re-oxidized by bromate forming temporal or spatial patterns of metal ion concentration in either oxidation state. This is a self-organized structure, because the reaction is not dominated by equilibrium thermodynamic behavior. The reaction is far from equilibrium and remains so for a significant length of time. Finally,... 

The phase coexistence observed around the first-order transition in NIPA gels cannot be interpreted by the Flory-Rehner theory because this theory tacitly assumes that the equilibrium state of a gel is always a homogeneous one. Heterogeneous structures such as two-phase coexistence are ruled out from the outset in this theory. Of course, if the observed phase coexistence is a transient phenomenon, it is beyond the thermodynamical theory. However, as will be described below, the result of the detailed experiment strongly indicates that the coexistence of phases is not a transient but rather a stable or metastable equilibrium phenomenon. At any rate, we will focus our attention in this article only on static equilibrium phenomena. 

The steady-state solution that is an extension of the equilibrium state, called the thermodynamic branch, is stable until the parameter A reaches the critical value A,. For values larger than A<, there appear two new branches (61) and (62). Each of the new branches is stable, but the extrapolation of the thermodynamic branch (a ) is unstable. Using the mathematical methods of bifurcation theory, one can determine the point A, and also obtain the new solution, (i.e., the dissipative structures) in the vicinity of A, as a function of (A - A,.). One must emphasize that... 

In examining a crystalline structure as revealed by diffraction experiments it is all too easy to view the crystal as a static entity and focus on what may be broadly termed attractive intermolecular interactions (dipole-dipole, hydrogen bonds, van der Waals etc., as detailed in Section 1.8) and neglect the actual mechanism by which a crystal is formed, i.e. the mechanism by which these interactions act to assemble the crystal from a non-equilibrium state in a super-saturated solution. However, it is very often nucleation phenomena that are ultimately responsible for the observed crystal structure and hence we were careful to draw a distinction between solution self-assembly and crystallisation at the beginning of this chapter. For example paracetamol, when crystallised from acetone solution gives the stable monoclinic crystal form I, but crystallisation from a molten sample in the absence of solvent... 

Note that y phase does not exist as a stable phase at the present temperature and pressure. In other words, if y phase is seen in the structure, then the system is not in die equilibrium, but a meta-stable, non-equilibrium state. However, if the thermodynamic state of the system is changed (e.g., different temperature or pressure), y phase may exist as a stable phase in a certain composition range ... 

There are two types of macroscopic structures equilibrium and dissipative ones. A perfect crystal, for example, represents an equilibrium structure, which is stable and does not exchange matter and energy with the environment. On the other hand, dissipative structures maintain their state by exchanging energy and matter constantly with environment. This continuous interaction enables the system to establish an ordered structure with lower entropy than that of equilibrium structure. For some time, it is believed that thermodynamics precludes the appearance of dissipative structures, such as spontaneous rhythms. However, thermodynamics can describe the possible state of a structure through the study of instabilities in nonequilibrium stationary states. 

Since the various 2- and 3-octyl cations will be in rapid equilibrium, the data in this Table are referenced to the most stable ground-state structure, cation 23. 

The experimentally observed manifestations of nonequdibricity of the stable catalyst state usually is less spectacular than those resulting in the for mation of dissipative structures. The most frequently observed manifesta tions of the stable nonequilibrium state are those in homogeneous systems, when the stationary concentrations of catalytic intermediates are substantially different in the course of the catalytic process and in the equilibrium system. 

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