  In a recent paper, we generalized a partition identity stated by Siladi\'c in
his study of the level one standard module of type $A_2^{(2)}$. The proof used
weighted words with an arbitrary number of primary colors and all the secondary
colors obtained from these primary colors, and a brand new variant of the
bijection of Bressoud for Schur's partition identity. In this paper, the first
of two, we analyze this variant of Bressoud's algorithm in the framework of
statistical mechanics, where an integer partition is viewed as an amount of
energy shared, according to certain properties, between several states. This
viewpoint allows us to generalize the previous result by considering a more
general family of minimal difference conditions. For example, we generalize the
Siladi\'c identity to overpartitions. In the second paper, we connect this
result to the Glaisher theorem and give some applications to level one perfect
crystals.
