  For classical discrete systems under constant composition, typically referred
to as substitutional alloys, we propose basic frameworks that provide
thermodynamic treatment of nonlinearity in canonical ensemble, i.e., nonlinear
correspondence between a set of many-body interaction and equilibrium
configuration through canonical average. We see that stochastic system
evolution driven by the nonlinearity itself can be naturally transformed into
system evolution contacting with a single thermal bath. This transformation
enables bridge two different concepts of the local nonlinearity on
configuration space and non-local nonlinearity on statistical manifold
previously introduced: We derive relationships between nonlinearity evolution
on statistical manifold and heat inflow through system evolution driven by the
nonlinearity on cofiguration space. Owing to the transformation, we find upper
bound for the average of the nonlinearity evolution, which is characterized by
entropy production of ideally linear system.
