  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.

Fluid Dynamics;Classical Physics