Equilibrium stationary

Remark. A great deal of attention has been paid in recent years to non-equilibrium stationary processes that are unstable and also extended in space. They can give rise to different phases that exist side by side, so that translation symmetry is broken. The name dissipative structures has been coined for them, and the prime examples are the Benard cells and the Zhabotinski reactions, but they also occur in biology and meteorology. However, these are features of the macroscopic equations. They are only relevant for fluctuation theory inasmuch as the fluctuation becomes very large at the point where the instability sets in. The critical fluctuations in XIII.5 are an example. There are many more varieties, in particular in the case of more variables. 

At equilibrium, stationary conditions exist where P(r,v t) = P(r,v) and P(r,v) is given by equilibrium statistical mechanics ... 

Table 1 shows that with the equilibrium stationary flow distribution that corresponds to the maximum entropy of an isolated system... 

Keywords Colloidal dispersion Flow curve Glass transition Integration through transients approach Linear viscoelasticity Mode coupling theory Nonlinear rheology Non-equilibrium stationary state Shear modulus Steady shear... 

Under shear, jj cannot vanish, as this would require the gradient term to balance the term proportional to y which, however, has a non-vanishing curl the potential conditions for an equilibrium stationary state are violated under shear [47],... 

Equilibrium stationary states i.e., the stationary states attained in a system in equilibrium. 

Stationarity follows from the fact that the time-correlation functions are defined as averages over equilibrium (stationary) ensembles. In such ensembles, it should not, and by Theorem C.l does not, matter what time is chosen as the initial time. Time correlation functions in stationary ensembles are invariant to a shift in the origin of time. 

Figure 6-18. Dependence of NO yield in non-equilibrium stationary plasma-beam discharge on electron beam power at gas flow rate 120 L/s and different levels of gas pressure (1) p = 5- 10 2 Torr (2) p = 0.2Torr.

Vukojevic, V., Pejic, N., Stanisavljev, D., Anic, S., and Kolar-Anic, Lj., Determination of Cl , Br , I , Mn " ", malonic acid and quercetin by perturbation of a non-equilibrium stationary state in the Bray-Liebhafsky reaction, Analyst, 124, 147-152, 1999. 

Example 6 Non-equilibrium stationary state as a characteristic of mechanochemical treatment Concurrent mechanochemical treatment of powder mixture of Bi203 and Xi02 in 2 3 molar ratio and pulverized Bi4Ti30i2 compound prepared by reactive sintering shows that after some milling time, a steady-state characterized by a very disordered, amorphous-like stmcture was reached. Thus, the systems evolves toward a non-equilibrium stationary state regardless of different initial thermodynamic states. 

In this book we offer a coherent presentation of thermodynamics far from, and near to, equilibrium. We establish a thermodynamics of irreversible processes far from and near to equilibrium, including chemical reactions, transport properties, energy transfer processes and electrochemical systems. The focus is on processes proceeding to, and in non-equilibrium stationary states in systems with multiple stationary states and in issues of relative stability of multiple stationary states. We seek and find state functions, dependent on the irreversible processes, with simple physical interpretations and present methods for their measurements that yield the work available from these processes. The emphasis is on the development of a theory based on variables that can be measured in experiments to test the theory. The state functions of the theory become identical to the well-known state functions of equilibrium thermodynamics when the processes approach the equilibrium state. The range of interest is put in the form of a series of questions at the end of this chapter. 

Systems not at equilibrium may be in a transient state proceeding towards equilibrium, or in a transient state proceeding to a non-equilibrium stationary state, or in yet more complicated dynamical states such as periodic oscillations of chemical species (limit cycles) or chaos. The first two conditions are well explained with an example consider the reaction sequence... 

Hence at the non-equilibrium stationary state, where by definition we have for the pressure of X at that state... 

For the transient relaxation of X to the non-equilibrium stationary state AG is not a valid criterion of irreversibility or spontaneous reaction. We shall develop necessary and sufEcient thermodynamic criteria for such cases. 

What are the thermodynamic functions that describe the approach of such systems to a non-equilibrium stationary state, both the approach of each intermediate species and the reaction as a whole ... 

What are the thermodynamic forces, conjugate fluxes and applicable extremum conditions for processes proceeding to or from non-equilibrium stationary states What is the dissipation for these processes ... 

Thus the stationary probability distribution of the master equation in the eikonal approximation is a Lyapunov function, which gives necessary and sufficient conditions of the existence and stability of non-equilibrium stationary states and provides a measure of relative stability on the basis of inhomogeneous fluctuations, (6.17). 

In Chap. 2 9 we presented a thermodynamic and stochastic theory of chemical reactions and transport processes in non-equilibrium stationary and transient states approaching non-equilibrium stationary states. We established a state function systems approaching equilibrimn reduces to AG. Since Gibbs free energy changes can be determined by macroscopic electrochemical measurements, we seek a parallel development for the determination of by macroscopic electrochemical and other measurements. 

Measurement of Electrochemical Potentials in Non-Equilibrium Stationary States... 

The net reaction is the oxidation of Ce(III) to Ce(IV) by bromate. In the bistable regime there is a state, where essentially no reaction occurs, which coexists with a state in which a percentage of Ce(III) is oxidized to Ce(IV). In this system we measured [6] at the same time the optical density which gives concentrations of Ce(IV) by Beer s law, and hence also the concentration of Ce(III) by conservation, and the emf of a Pt electrode which at equilibrium follows the Nernst equation (10.1). The experiment consisted of the measurement of the emf of the Ce(III)/Ge(IV) half reaction at a redox (Pt-Ag/AgGl) electrode imder equilibrium and stationary non-equilibrium conditions. The apparatus is shown in Fig. 10.1, but in these experiments the parts 4 7 were not present. From these measurements we determined that there exists a non-Nernstian contribution in a non-equilibrium stationary state as shown in Table 10.2. The concentration of [Ce(III)]ss in the stationary state is obtained... 

A non-equilibrium stationary state is achieved by flowing the reacting solutions into the CSTR at given flow rates, that is given residence times in the reactor the measurements just described are repeated, and shown for a residence time of 175 s in Fig. 10.3, and a residence time of 400 s in Fig. 10.4. 

First, it is interesting to compare the equilibrium displacement plot (Fig. 10.2) with the plots of displacements from non-equilibrium stationary... 

Fig. 10.3. Experiments as in Fig. 10.2 for a non-equilibrium stationary state at zero imposed current and displacements from that state with imposed currents. The residence time is 175 s. The arrows indicate transitions to other stationary states,... 

Consider the chemical system in (11.1) with the species being either ions or neutrals the system is in a reaction chamber in a non-equilibrium stationary state. We impose a flux of species Ai, J = k A[, into the reaction chamber with <5+ and Q- held constant and thereby move the chemical system to a different non-equilibrium stationary state with different concentrations of the reacting species Ai, Bi, A, B2- This procedure allows the sampling of different combinations of the reacting species by means of the imposition of different fluxes of these reactants. These combinations represent different non-stationary states in the absence of imposed fluxes, but with the imposed fluxes they are stationary states and hence measurements may be made without constraints of time. If we would attempt to measure concentrations in non-stationary states then the measurement technique would have to be fast compared to the time scale of change of the concentrations due to chemical reactions. 

A direct test of the master equation for systems in non-equilibrium stationary states comes from the measurements of concentration fluctuations such measurements have not been made yet. Some other tests of the master equation are possible based on the earlier sections in this chapter, where we can compare measurements of the stochastic potential with numerical solutions of the master equation (which requires knowledge of rate coefficients and the reaction mechanism of the system). 

Thermodynamics is one of the foundations of science. The subject has been developed for systems at equilibrium for the past 150 years. The story is different for systems not at equilibrium, either time-dependent systems or systems in non-equilibrium stationary states here much less has been done, even though the need for this subject has much wider applicability. We have been interested in, and studied, systems far from equilibrium for 40 years and present here some aspects of theory and experiments on three topics ... 

Thus the term d/dt( S) = Zm dFu dju > 0 (simplified at the nearequilibrium conditions as ZmFmJm> 0) must be positive if the non-equilibrium stationary state is stable. 

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