On the Role of the Integrable Toda Model in One-Dimensional Molecular Dynamics

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.