On the Formalism of Thermodynamic Fluctuation Theory

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Abstract
From the statistical mechanical occupation probability of microstates Einstein has inferred a distribution function for the macroscopic states of a canonical ensemble. The conventional theory of thermodynamic fluctuations proceeds by making certain series expansions in the Einstein function and by dropping all cubic and higher order terms. In this paper we establish that: (a) The correlation moments for the extensive thermodynamic parameter fluctuations may be computed directly from the distribution function for the microstates, without introducing an intermediate macroscopic distribution function. (b) These same moments can be evaluated from the Einstein function without making series expansions or invoking approximations. (c) All moments computed by methods (a) or (b) agree exactly. (This may be taken as an alternative derivation of the Einstein function.) (d) The second moments computed by the conventional method are correct, but all higher moments are incorrect.

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