A geometrical approach to computing free energy landscapes from short-ranged potentials

TL;DR

This paper introduces a geometrical framework for analyzing free energy landscapes of particles with short-ranged potentials, focusing on the zero-range limit where landscapes are defined by geometric manifolds independent of potential details.

Contribution

It develops a novel geometrical approach to characterize energy landscapes in the zero-range limit, linking them to fundamental manifolds and providing a method to compute dynamical quantities.

Findings

Manifolds are independent of potential details.

Low-dimensional manifolds for up to 8 particles characterized.

Asymptotic Fokker-Planck dynamics restricted to these manifolds.

Abstract

Particles interacting with short-ranged potentials have attracted increasing interest, partly for their ability to model mesoscale systems such as colloids interacting via DNA or depletion. We consider the free energy landscape of such systems as the range of the potential goes to zero. In this limit, the landscape is entirely defined by geometrical manifolds, plus a single control parameter. These manifolds are fundamental objects that do not depend on the details of the interaction potential, and provide the starting point from which any quantity characterizing the system -- equilibrium or non-equilibrium -- can be computed for arbitrary potentials. To consider dynamical quantities we compute the asymptotic limit of the Fokker-Planck equation, and show that it becomes restricted to the low-dimensional manifolds connected by "sticky" boundary conditions. To illustrate our theory, we compute the low-dimensional manifolds for n<=8 identical particles, providing a complete description of the lowest-energy parts of the landscape including floppy modes with up to 2 internal degrees of freedom. The results can be directly tested on colloidal clusters. This limit is a novel approach for understanding energy landscapes, and our hope is that it can also provide insight into finite-range potentials.