Equilibrium statistical mechanics postulates

The structure of the chapter is as follows. In Section 2, we review the basic postulates of the equilibrium thermodynamics. The equilibrium statistical mechanics based on generalized entropy is formulated in a general form in Section 3. In Section 4, we describe the Tsallis statistics and analyze its possible connection with the equilibrium thermodynamics. The main conclusions are summarized in the final section. 

The two postulates of equilibrium statistical mechanics introduced in Chapter 25 can now be restated ... 

Since we can never wait infinitely long time (Xexp cxd) to actually observe a SM S, its existence can never be verified. This is no different from what is customary in equilibrium statistical mechanics, where the existence of the equilibrium state is taken for granted as a postulate. We quote Huang ([41], p. 127) Statistical mechanics, however, does not describe how a system approaches equilibrium, nor does it determine whether a system can ever be found to be in equilibrium. It merely states what the equilibrium situation is for a given system. Ruelle ([42], p. 1) notes that equilibrium states are defined operationally by assuming that the state of an isolated system tends to an equilibrium state as time tends to + oo. Whether a real system actually approaches this state cannot be answered. Therefore, we will... 

If we interpret Gibbs entropy in the same spirit as the missing information of information theory, it can be viewed as a measure of statistical uncertainty. Adopting this point of view, it seems natural to treat the following principle as a basic postulate of classical equilibrium statistical mechanics. 

Various statistical treatments of reaction kinetics provide a physical picture for the underlying molecular basis for Arrhenius temperature dependence. One of the most common approaches is Eyring transition state theory, which postulates a thermal equilibrium between reactants and the transition state. Applying statistical mechanical methods to this equilibrium and to the inherent rate of activated molecules transiting the barrier leads to the Eyring equation (Eq. 10.3), where k is the Boltzmann constant, h is the Planck s constant, and AG is the relative free energy of the transition state [note Eq. (10.3) ignores a transmission factor, which is normally 1, in the preexponential term]. 

Considerable effort has been expended in the attempt to develop a general theory of reaction rates through some extension of thermodynamics or statistical mechanics. Since neither of these sciences can, by themselves, yield any information about rates of reactions, some additional assumptions or postulates must be introduced. An important method of treating systems that are not in equilibrium has acquired the title of irreversible thermodynamics. Irreversible thermodynamics can be applied to those systems that are not too far from equilibrium. The theory is based on the thermodynamic principle that in every irreversible process, that is, in every process proceeding at a finite rate, entropy is created. This principle is used together with the fact that the entropy of an isolated system is a maximum at equilibrium, and with the principle of microscopic reversibility. The additional assumption involved is that systems that are slightly removed from equilibrium may be described statistically in much the same way as systems in equilibrium. 

The theory of absolute reaction rates, which i s based on statistical mechanics, was developed in full generality by H. Eyring in 1935, although it was foreshadowed in kinetic theory investigations as early as 1915. A simplified development of the equations will be given here. In this theory, we have a postulate of equilibrium away from equilibrium, applied more broadly here than in the irreversible thermodynamics. 

The third law of thermodynamics, like the first and second laws, is a postulate based on a large number of experiments. In this chapter we present the formulation of the third law and discuss the causes of a number of apparent deviations from this law. The foundations of the third law are firmly rooted in molecular theory, and the apparent deviations from this law can be easily explained using statistical mechanical considerations. The third law of thermodynamics is used primarily for the determination of entropy constants which, combined with thermochemical data, permit the calculation of equilibrium constants. 

The main postulate of statistical mechanics states that, for an equilibrium system of given mass, composition, and spatial extent, all microstates with the same energy are equally probable [37,38]. This postulate, along with the ergodic hypothesis, can be justified on the basis of the mixing flow in phase space exhibited by the dynamical trajectories of real systems [28]. It means that all microstates of an isolated system, which does not exchange energy and material with its environment, should occur equally often. 

The dassical mechanics of the first three sections of Chapter 1, the riaiatical thermodynamics of Chapter 2, and the statistical mechanics of the first four sections of this chapter all have one feature in common— the discussion is strictly confined to equilibrium states of matter that can, at least in principle, be studied in the laboratory. The first breadi in this position of self-restraint came when J. Hiomson (1871) and van der Waals (1873) suggested that Andrews s experiments on the continuity of state made it reasonable to discuss the properties of fiuids whose densities were between those of the orthobaric gas and liquid. These states played a key role in the development of the theories of surface tension of Rayleigh (1892) and, more important, of van der Waals (1893). The later quasi-thermodynamic work of Chapter 3 shows how fruitful it has been to postulate the existence of thermodynamic functions sudi as the free energy for values of their arguments other than those that describe the state of equilibrium. 

In the above formal development, we found that Gibbs stability condition from equilibrium thermodynamics and Prigogine s stability condition from nonequilibrium thermodynamics for a chemically reactive system emerge from the statistical mechanical treatment of nonequilibrium systems. Unlike the stability conditions of Gibbs and Prigogine, the inequalities (384), (385), (396), and (397) are not postulates. They are simple consequences of the statistical mechanical treatment. Moreover, these inequalities apply to both equihbrium and nonequilibrium systems. 

The simple statistical mechanical models, i.e. affine and phantom, or the phenomenological approach cannot reproduce the maximum in the plot of Aln(ai/a ) vs. n6,28i jjjjg maximum is predicted, however, by the constrained junction theory but, as shown in Figure 19, agreement with experiment is only qualitative.This has to lead to questioning of the separability of mixing and elastic contributions in equation (145). This postulate may be valid only in the limit of swelling equilibrium. ... 

Several theorems that can be derived from the three postulates of quantum mechanics named above have been presented In the literature. One of these is that to every state of a system specified by means of a given preparation there corresponds a Her-mitian operator (3, called the density operator, which is an index of measurement statistics. The incorporation of the stable-equilibrium postulate into the theory, however, gives rise to additional theorems that are new to quantum physics. Some of these new theorems are as follows ... 

We resort rather to a postulate, in addition to those of mechanics, which is of statistical character and asserts that For a system in internal equilibrium the probability of a given quantum state is a function of its energy only. 

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