Lagrangian dynamics of thermal tracer particles in Navier-Stokes fluids

A key issue in fluid dynamics is the definition of the phase-space Lagrangian dynamics characterizing prescribed ideal fluids (i.e., continua), which is related to the dynamics of so-called \textit{ideal tracer particles} moving in the same fluids. These are by definition particles of infinitesimal size which do not produce significant perturbations of the fluid fields and do not interact among themselves. In this work we point out that the phase-space Lagrangian description of incompressible Navier-Stokes fluids can be achieved by means of a particular subset of ideal tracer particles, denoted as thermal particles. For these particles the magnitude of their relative velocities - with respect to the local fluid velocity - is solely determined by the kinetic pressure, in turn, uniquely related to the fluid pressure. The dynamics of thermal tracer particles is shown to generate the time-evolution of the fluid fields by means of a suitable statistical model. The result is reached introducing a 1-D statistical description of the fluid exclusively based on the ensemble of thermal tracer particles. In particular, it is proven that the statistic of thermal particles can be uniquely defined requiring that for these particles the directions of their initial relative velocities are defined by a suitable family of random coplanar unit vectors.