  For classical systems, expectation value of macroscopic property in
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 cofiguration
space). Although we have a number of theoretical approaches enabling to
quantitatively estimate equilibrium properties by applying given potential
energy surface (PES) to the thermodynamic average, it is generally unclear
whether PES can be stablly, inversely determined from a given set of
properties. 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 classical discrete system,
this property for a set of microscopic states satisfying special condition can
be well-characterized by a newly-introduced concept of anharmonicity in the
structural degree of freedom of D, where these states are expected to be stably
inversed to underlying PES, 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: We find that the breaking is mainly dominated by the sum of
divergence and Jacobian of vector field D in configuration space, which can be
fully known a priori only from geometric information of underlying lattice,
without using any thermodynamic information such as energy or temperature.
