On the role of the Integrable Toda model in one-dimensional molecular dynamics

TL;DR

This paper demonstrates that molecular potentials like Mie-Lennard-Jones, when normalized, approximate the integrable Toda model in the limit of strong repulsion, revealing near-integrable dynamics in typical molecular chains.

Contribution

It establishes a theoretical link between common molecular potentials and the Toda integrable system, supported by numerical evidence showing closeness to integrability.

Findings

MLJ potentials converge to Toda potential in the hard-core limit

FPU-like Hamiltonians with MLJ potentials are close to Toda Hamiltonian

Numerical results show standard MLJ potential is closer to integrability than FPU potentials

Abstract

We prove that the common Mie-Lennard-Jones (MLJ) molecular potentials, appropriately normalized via an affine transformation, converge, in the limit of hard-core repulsion, to the Toda exponential potential. Correspondingly, any Fermi-Pasta-Ulam (FPU)-like Hamiltonian, with MLJ-type interparticle potential, turns out to be $1/n$-close to the Toda integrable Hamiltonian, $n$ being the exponent ruling repulsion in the MLJ potential. This means that the dynamics of chains of particles interacting through typical molecular potentials, is close to integrable in an unexpected sense. Theoretical results are accompanied by a numerical illustration; numerics shows, in particular, that even the very standard 12--6 MLJ potential is closer to integrability than the FPU potentials which are more commonly used in the literature.