Microscopic Geometry Characterizes Structure/Potential-Energy Correspondence in a Thermodynamic System

Abstract

Potential energy landscape (PEL) is essential to determine phase stability, reaction path, and other important physical as well as chemical properties. Whereas given PEL can reasonably determine the properties in thermodynamically equilibrium state, it is generally unclear whether a set of known property can uniquely and/or stably determines PEL, i.e., understandings of property/PEL correspondence is basically unidirectional in the current statistical mechanics. Here we make significant advance toward bidirectional bridging of this gap for classical discrete systems under many-body interactions. Our idea is to focus on characteristic microscopic geometry in configuration space for an exactly solvable system, resulting in a new, important quantity of "harmonicity in the structural degree of freedom". This quantity reasonablly characterizes which structures in equilibrium state have practically unique and stable correspondence to PEL, without requiring any thermodynamic information such as energy or temperature. The present findings will open a gate to constructing reliable PEL, where its predictive uncertainty can be a priori known. A significant role of microscopic geometry for non-interacting system should be re-emphasized in statistical mechanics.

Universal Characterization of Hierarchical Ordering Tendency in High-Entropy Alloys from Configurational Geometry

“…We can clearly see that (i) conventional Goldschmidt atomic radius cannot capture ordering tendency for whole or any subsystems, (ii) R from unary system has weak correlation with SRO for whole and subsystems due mainly to the poor information about atomic radius in preferring ordering configuration and thus to the poor variety of resultant atomic radius, and (iii) R from projection state, i.e., explicitly including covariance fluctuation between structural degree of freedoms in configurational geometry, 13 can universally well-characterize ordering tendency for whole as well as subsystems. Therefore, we then marginally plot α 4 , α 3 and α 2 with atomic radius ratio R PS , shown in Fig.…”

Thermodynamics for Nonlinearity under Hidden Structure Information

“…Our recent theoretical study reveal that nonlinearity can be reasonablly treated through introduced vector field A of anharmonicity in s.d.f. (ASDF) depending only on configurational geometry, 5,6 where A can be naturally treated as time-evolution of discrete dynamical system. We quantitatively formulate bidirectional stability (BS) character of thermodynamic average between equilibrium structure and poten-tial energy surface in terms of their hypervolume correspondence, by divergence and Jacobian for A, 7 and we examine the origin of nonliearity based on tropical geometry and information geometry with dually flat Riemannian manifold, which clarifies how spatial constraint to individual s.d.f., entanglement between s.d.f.…”

Thermodynamics for Nonlinearity in Canonical Ensemble

“…Our recent theoretical study reveal that nonlinearity can be reasonablly treated through introduced vector field A of anharmonicity in s.d.f. (ASDF) depending only on configurational geometry, 5,6 where A can be naturally treated as time-evolution of discrete dynamical system. We quantitatively formulate bidirectional stability (BS) character of thermodynamic average between equilibrium structure and potential energy surface in terms of their hypervolume correspondence, by divergence and Jacobian for A, 7 and we further examine the origin of nonliearity based on tropical geometry and information geometry with dually flat Riemannian manifold, which clarifies how spatial constraint to individual s.d.f., entanglement between s.d.f.…”