  The principle of Random Galilean Transformation (RGT) Invariance is applied
to the random motion of particles in a turbulent gas to construct a kinetic
equation for the transport of the particle phase space probability
<W(\mathbf{v},\mathbf{x},t)> where \mathbf{v} and \mathbf{x} are the velocity
and position of a particle at time t. The essential problem is to find closed
expressions for the phase space dispersion current \left\langle
\boldsymbol{f}W\right\rangle , where \boldsymbol{f} is the fluctuating
aerodynamic force at \mathbf{v} and \mathbf{x} at time t. The simplest form
consistent with RGT invariance, the correct equation of state ancl form for the
inter-phase momentum transfer tern is shown to be \left\langle
\boldsymbol{f}W\right\rangle
=-\left(\boldsymbol{\mu}\cdot\frac{\partial}{\partial\mathbf{v}}+\boldsymbol{\lambda}\cdot\frac{\partial}{\partial\mathbf{x}}\right)\left\langle
W\right\rangle in which
\text{\textbf{\ensuremath{\boldsymbol{\,\mu}}}\ensuremath{=<\boldsymbol{f}(t)\mathbf{v}(t)>}}
and \boldsymbol{\lambda}=<\boldsymbol{f}(t)\boldsymbol{\mathbf{x}}(t)>.This
approach to modeling gas-solid flows is currently being used to investigate the
behavior of radioactive aerosols inside gas-cooled nuclear reactors.
