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

## TL;DR

This paper introduces a microscopic geometric measure called 'harmonicity' to characterize the stable and unique correspondence between structure and potential energy landscape in classical systems, advancing the understanding of property-PEL relationships.

## Contribution

It proposes a new quantity, harmonicity, based on microscopic geometry, to determine when structures have a stable, unique PEL correspondence without thermodynamic data.

## Key findings

- Harmonicity characterizes structure-PEL correspondence stability.
- The approach applies to exactly solvable classical systems.
- It enables constructing reliable PEL with known predictive uncertainty.

## 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.