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Tropical Limit for Configurational Geometry in Discrete Thermodynamic Systems

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.