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

Abstract

This is a purely analytical information theory work. We will uncover interesting links between Fisher’s measure and Gibbs’ ensembles, particulary the grand canonical and canonical ones, each offering distinct insights into a system’s properties. In this effort, we construct an informational connection between these ensembles, based on Fisher’s measure. We specifically explore the relationships among four key concepts: • The canonical and grand canonical ensembles. • Fisher information measures applied to three parameters (inverse temperature, particle number, and fugacity). • The Fano factor. • Special Poisson probability distributions (PDs) that are related to grand canonical PDs by having the same mean particle number ⟨N⟩. Our findings reveal intricate and compelling relationships among these concepts, providing new perspectives on the interplay between statistical mechanics and information theory.