  For classical discrete systems with constant composition (typically referred
to substitutional alloys) under thermodynamically equilibrium state,
macroscopic structure should in principle depend on temperature and many-body
interaction through Boltzmann factor, exp(-bE). Despite this fact, our recently
find that (i) thermodynamic average for structure can be characterized by a set
of special microscopic state whose structure is independent of energy and
temperature, and (ii) bidirectional-stability character for thermodynamic
average between microscopic structure and potential energy surface is
formulated without any information about temperature or many-body interaction.
These results strongly indicates the significant role of configurational
geometry, where anharmonicity in structural degree of freedom (ASDF) that is a
vector field on configuration space, plays central role, intuitively
corresponding to nonlinearity in thermodynamic average depending only on
configurational geometry. Although ASDF can be practically drawn by performing
numerical simulation based on such as Monte Carlo simulation, it is still
unclear how its entire character is dominated by geometry of underlying
lattice. We here show that by applying the limit in tropical geometry (i.e.,
tropical limit) with special scale-transformation to discrete dynamical system
for ASDF, we find tropical relationships between how nonlinearity in terms of
configurational geometry expands or shrinks and geometric information about
underlying lattice for binary system with a single structural degree of freedom
(SDF). The proposed tropical limit will be powerful procedure to simplify
analyzation of complicated nonlinear character for thermodynamic average with
multiple SDF systems.
