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On the basis of the results obtained in a previous paper it is shown that in the thermodynamic limit the analogues of the Massieu-Plandc functions are linked with each other by means of the Legendre transformation. The existence of the limiting function φk(∞) implies the existence of the limiting function φl(∞) (l<k) under the same assumptions. Passage to the limit and derivation with respect to all independent variables commute. A statistical derivation of the thermodynamic stability condition in its most general form is given which leads naturally to a statistical interpretation of the concept of thermodynamic stability.
It is shown that, for all conceivable ensembles of statistical thermodynamics, at the thermodynamic limit, the frequency function of the fluctuations of macroscopic extensive parameters equals a Gaussian. The proof is based on a generalisation of Khinchin's method using the concept of "smoothed frequency functions."