Saddles in the energy landscape: extensivity and thermodynamic formalism

TL;DR

This paper extends the energy landscape framework to include saddle points, deriving their scaling with system size and formulating a partition function that captures temperature-dependent properties of saddles in liquids.

Contribution

It introduces a rigorous energy landscape approach that accounts for saddle points and reproduces observed simulation trends in liquid thermodynamics.

Findings

Saddles scale extensively with system size.

Partition function explicitly includes saddle contributions.

Temperature dependence of saddle energy and order is accurately modeled.

Abstract

We formally extend the energy landscape approach for the thermodynamics of liquids to account for saddle points. By considering the extensive nature of macroscopic potential energies, we derive the scaling behavior of saddles with system size, as well as several approximations for the properties of low-order saddles (i.e., those with only a few unstable directions). We then cast the canonical partition function in a saddle-explicit form and develop, for the first time, a rigorous energy landscape approach capable of reproducing trends observed in simulations, in particular the temperature dependence of the energy and fractional order of sampled saddles.