Source of mode coupling via projection between momentum and coordinate-defined unit vectors

Establish whether, in the coarse-grained dynamics of a suspended single-walled carbon nanotube modeled with two Hamiltonians for bond-length and angle modes, the mode coupling mechanism arises from the cross-correlation projection of momentum in one mode onto the unit vector defined by the coordinates of the other mode at the next time step; specifically, determine if the source of mode coupling is the term formed by projecting P_i e_i(t) onto e_j(t+Δt), where P_i is the momentum amplitude in mode i and e_j(t+Δt) is the coordinate-defined unit vector of mode j at time t+Δt.

Background

The paper models a single-walled carbon nanotube (SWCNT) as a coarse-grained system with two Hamiltonians representing bond-length and angle degrees of freedom. Due to time-evolving local coordinate frames, momentum components associated with one Hamiltonian can project onto the coordinate-defined unit vectors of the other, producing cross-correlation effects described as a Dzhanibekov-like phenomenon. The authors derive an evolution equation using Smoluchowski dynamics and propose a perturbation energy φ to account for these cross-correlations.

In the Discussion (Collision response), the authors compare molecular dynamics simulations and coarse-grained simulations and suggest that mode coupling may originate from a specific projection term linking momentum and coordinate-defined unit vectors across successive time steps. They label this as a conjecture, indicating the mechanism is not yet established.