Information Theory’s Links Between Fisher Measure and Gibbs’ Ensembles

This is an information theory effort. We discover interesting links between Fisher measure and Gibbs’ ensembles. Among these ensembles, the grand canonical and canonical ensembles are fundamental, each offering distinct insights into a system’s properties. In this work, we construct a deep connection between these ensembles, generated by Fisher’s information measure. We specifically explore the relationships among four key concepts: (1) the canonical and grand canonical ensembles, (2) continuous and discrete Fisher information measures applied to three parameters: (a) inverse temperature, (b) particle number, and (c) fugacity, and (3) the Fano factor, and (4) specialized Poisson distributions defined by the particle number and its mean value. Our findings reveal intricate and compelling relationships among these concepts, providing new perspectives on the interplay between statistical mechanics and information theory.