Bidirectional-Stability Breaking in Thermodynamic Average for Classical Discrete Systems

For classical systems, expectation value of macroscopic structure in thermodynamically equilibrium state can be typically provided through thermodynamic (so-called canonical) average, where summation is taken over possible states in phase space (or in crystalline solids, it is typically approximated on configuration space). Although we have a number of theoretical approaches enabling to quantitatively estimate equilibrium structures by applying given potential energy surface (PES) to the thermodynamic average, it is generally unclear whether PES can be stably, inversely determined from a given structure. This essentially comes from the fact that bidirectional stability characters of thermodynamic average for classical system is not sufficiently understood so far. Our recent study reveals that for substitutional alloys typically referred to classical discrete system under constant composition, this property can be qualitatively well-characterized by a newly-introduced concept of "anharmonicity in the structural degree of freedom" of D, known without any thermodynamic information. However, it is still quantitatively unclear how the bidirectional stability character is broken inside the configuration space. Here we show that the breaking in bidirectional stability for thermodynamic average is quantitatively formulated from information only about configurational geometry: We find that the breaking is provided by the sum of divergence and Jacobian of vector field D in configuration space fully known a priori without any thermodynamic information, which indicates that nonlinearity of thermodynamic average in terms of configurational geometry should play essential role to determine the bidirectional stability character.