Ensemble transformation in the fluctuation theory

Interactions in complex solutions that consist of multiple components can be quantified via number correlations observed within an isochoric subsystem. The fluctuation solution theory carries out a conversion between experimental data (usually measured under isobaric conditions) and the isochoric number correlations through cumbersome thermodynamic variable transformations. In contrast, we have recently demonstrated heuristically that direct transformation of statistical variables (i.e., those variables fluctuating in statistical ensembles such as volume and particle numbers) can lead to equivalent results via simple algebra. This paper reveals the geometrical basis of fluctuation and the invariants underlying the equivalence between thermodynamic and statistical variable transformations. Based on the quasi-thermodynamic fluctuation theory and the postulate that concentration and its fluctuation are invariant under ensemble transformation, we show that the thermodynamic and statistical variable transformations correspond to the change of basis on the Hessian matrix and the vectors whose elements are the deviations of statistical variables from their mean values, respectively, under which the quadratic form of fluctuation is invariant. Statistical variable transformation can also be used in cases when a set of experimental data do not belong to the same ensemble. When combined with the order-of-magnitude analysis, our formalism shows that the quasi-thermodynamic formalism of fluctuation at the thermodynamic limit is valid only for extensive variables and cannot be applied to intensive variables.