Statistical thermodynamics small deviations

Let us consider the compounds which show a small deviation from the stoichiometric composition and whose non-stoichiometry is derived from metal vacancies. The free energy of these compounds, which take the composition MX in the ideal or non-defect state, can be calculated by the method proposed by Libowitz. To readers who are well acquainted with the Fowler-Guggenheim style of statistical thermodynamics, the method here adopted may not be quite satisfactory however, the Libowitz method is understandable even to beginners who know only elementary thermodynamics and statistical mechanics. It goes without saying that the result calculated by the Libowitz method is essentially coincident with that calculated by the Fowler-Guggenheim method. 

In this chapter, we first present some of the notation that we shall use throughout the book. Then we summarize the most important relationship between the various partition functions and thermodynamic functions. We shall also present some fundamental results for an ideal-gas system and small deviations from ideal gases. These are classical results which can be found in any textbook on statistical thermodynamics. Therefore, we shall be very brief. Some suggested references on thermodynamics and statistical mechanics are given at the end of the chapter. 

If we look at a small portion of a macroscopic system or study a mesoscopic system, we must study fluctuations. The probability of fluctuations is phenomenologically described by the thermodynamic theory of fluctuations (5). From the ensemble theory point of view, the fluctuation theory is the study of large deviations from the expectation value. This is the reason why large deviation theory is becoming increasingly important in statistical thermodynamics. Standard works on large deviation theory are References 16 and 17 perhaps as accessible introduction to the topic may be found in Reference 18. 

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