Tropical Limit for Configurational Geometry in Discrete Thermodynamic Systems

For substitutional alloys typically referred to classical discrete systems with constant composition under thermodynamically equilibrium state, macroscopic structure should in principle depend on temperature and many-body interaction through Boltzmann factor, exp(-beta E). Despite this fact, our recent study finds that (i) thermodynamic average for structure can be characterized by a set of special microscopic state whose structure is independent of energy and temperature [K. Yuge, J. Phys. Soc. Jpn. 85, 024802 (2016)], 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 [K. Yuge and S. Ohta, J. Phys. Soc. Jpn. 88, 104803 (2019)]. These results strongly indicate 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 suitable limit in tropical geometry to discrete dynamical system for ASDF, we find significant 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 approach will be powerful procedure to simplify analyzation of complicated nonlinear character for thermodynamic average with multiple SDF systems.