A note on the encapsulated dumb-bell model (Q1080295)

There are at least two approaches to the kinetic theory of bead-spring molecules. In the first approach one writes the equations of motion of a bead-spring molecule subjected to a Brownian force which is modelled by additive white noise of a strength dictated by a fluctuation-dissipation theorem. In the second approach, one appeals to a rapid establishment of the ''Maxwellian distribution'' in the phase space of the molecule. We wish to show that the two approaches are consistent provided that the ''Maxwellian distribution'' is derived from the phase-space distribution equation of the system. Second, we wish to point out that in the recently-published dumb-bell model for concentrated polymer solutions [(*) \textit{R. B. Bird} and \textit{J. R. DeAugiar}, ibid. 13, 149-160 (1983; Zbl 0547.76002)] the expression for the ''smoothed out'' Brownian force is an additional assumption, because one does not know the correct ''Maxwellian'' distribution for an encapsulated dumb-bell molecule. However, if one views the system as a dilute suspension of ''encapsulated'' dumb-bells, each experiences the same spring force and anisotropic Stokes' drag as adopted in (*), then the expression for the ''smoothed out'' Brownian force, the diffusion equation and the expression for the stress tensor are not correct, if the ''encapsulated'' dumb-bell is to fully participate in the thermodynamics of the system.