  We study the non-equilibrium thermodynamics of single Brownian macromolecules
immersed in water solvent. They are under both a hydrodynamic interaction and a
feedback control on their movement by an external agent. The macromolecules are
described by a Langevin equation with a multiplicative noise. Work done by the
macromolecules on the water solvent is dissipated as heat. Thus, the heat is
expressed as the integration of an interacting force between the macromolecules
and the water solvent along the position space trajectories of the
macromolecules. This integration is stochastic due to the Brownian motion of
the macromolecules. We show that the Stratonovich prescription of the
integration is the unique physical choice. We also show that thermodynamic
quantities such as heat, work, and entropy production, are derived without any
ambiguity if both a diffusion matrix and external feedback control are known as
priori.
