  When many body system in Classical Mechanics is described using
bi-Hamiltonian for its group of particles which forms a coarse grained bead,
the damping energy from the heat bath perturbs the motion of a particle which
is along the trajectory of multiple Hamiltonian system simultaneously. As
managing the coarse grained modeling of the suspended semiflexible polymer like
single walled carbon nanotubes(SWCNT) in vacuum, the authors discovered that
such interconnection between Hamiltonian systems as sharing a heat bath builds
a strongly coupled system. The consequence of this interconnection is revealed
to form certain form of cross correlated momentum in coarsed grained system as
well as in the original atomic scales. In this paper, we adapt the framework of
Stochastic Thermodynamics to describe the cross correlation of bi-Hamiltonian
system which replicates the macroscopic behaviour of SWCNT under the
Smoluchowski picture for cross correlation between Hamiltonian systems. The
numerical experiments using collision between tubes confirms the derivation and
the justification of the usage of the heat diffusion to compensate the
abnormaly from cross correlated momentum.
