Statistical thermodynamics generalized postulates

Thermodynamics is a simple, general, logical science, based on two postulates, the first and second laws of thermodynamics. We have seen in the last chapter how to derive results from these laws, though we have not used them yet in our applications. But we have seen that they are limited. Typical results are like Eq. (5.2) in Chap. II, giving the difference of specific heats of any substance, CP — CV in terms of derivatives which can be found from the equation of state. Thermodynamics can give relations, but it cannot derive the specific heat or equation of state directly. To do that, we must go to the statistical or kinetic methods. Even the second law is simply a postulate, verified because it leads to correct results, but not derived from simpler mechanical principles as far as thermodynamics is concerned. We shall now take up the statistical method, showing how it can lead not only to the equation of state and specific heat, but to an understanding of the second law as well. 

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. 

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. 

Unlike other branches of physics, thermodynamics in its standard postulation approach [272] does not provide direct numerical predictions. For example, it does not evaluate the specific heat or compressibility of a system, instead, it predicts that apparently unrelated quantities are equal, such as (1 A"XdQ/dP)T = - (dV/dT)P or that two coupled irreversible processes satisfy the Onsager reciprocity theorem (L 2 L2O under a linear optimization [153]. Recent development in both the many-body and field theories towards the interpretation of phase transitions and the general theory of symmetry can provide another plausible attitude applicable to a new conceptual basis of thermodynamics, in the middle of Seventies Cullen suggested that thermodynamics is the study of those properties of macroscopic matter that follows from the symmetry properties of physical laws, mediated through the statistics of large systems [273], It is an expedient happenstance that a conventional simple systems , often exemplified in elementary thermodynamics, have one prototype of each of the three characteristic classes of thermodynamic coordinates, i.e., (i) coordinates conserved by the continuous space-time symmetries (internal energy, U), (ii) coordinates conserved by other symmetry principles (mole number, N) and (iii) non-conserved (so called broken ) symmetry coordinates (volume, V). 

In general we may, however, say that the situation in the field of statistical mechanics of multicomponent systems is more favourable than for simple components. For the latter, there are only a few cases where there exists a satisfactory theory which is able to predict the thermodynamic properties of the system from first principles (imperfect gas, two dimensional Ising model...). New basic ideas are necessary for progress. In the theory of multicomponent systems it is, however, possible to gain at least a partial success by aiming for a less ambitious goal. If we postulate that the solution to the problem of single-component systems is known, we may try to express the properties of multicomponent systenos as far as possible in terms of those of pure components. 

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