  The Legendre transformation is a crucial tool in theoretical physics, known
for its symmetry, especially when applied to multivariate functions. In
statistical mechanics, ensembles represent the central focus. Leveraging the
dimensionless aspect of Legendre transformation, this paper explores the
transformation process from the entropy characteristic function of
microcanonical ensembles to the analogous definition of partition function
transformation. Additionally, it derives characteristic functions, partition
functions, and establishes their interrelations, along with deriving
corresponding thermodynamic formulas for various ensembles. This streamlined
approach sheds light on the fundamental principles of statistical mechanics and
underscores the symmetry inherent in Legendre transformation.
