  The Langevin Equation for cooperative dynamics represents the dynamics of
polymer melts with chains of increasing degree of polymerization, covering the
full range of behavior from the unentangled to the entangled regime. This
equation describes the motion of a group of interpenetrating polymers that are
interacting through an effective potential resulting from the many-body
coupling of the inter polymer potential inside the macromolecular liquid. The
confinement of the dynamics due to the presence of entanglements is accounted
for by an effective inter-monomer potential which is zero until the distance
between two monomers belonging to different chains reaches a characteristic
value, d. At that distance a constraint is applied through an effective
hard-core repulsion that represents the effect of entanglements in the slowing
down of the relative diffusion of the monomers. As the time evolves the
constraint relaxes due to the chain interdiffusion. The same potential acts on
both unentangled and entangled polymer chains, but short chains are not
affected as they relax faster than they experience the presence of the
potential. The comparison with Neutron Spin Echo experiments of polyethylene
melts shows quantitative agreement of the theory with the time dependent
incoherent scattering function for a variety of samples in both the unentangled
and the entangled regimes.
