by a solid particle in a time $\tau_S$: $u_S\sim S_c/\tau_S$; the size $S_i$ of an eddy whose lifetime equals the crossing time by a solid particle: $S_i\gamma_{S_i^{-1}}\sim u_S$. Summarizing: $$S=\rho^\frac{3}{2}\Ckol^\frac{3}{4}\flux^\frac{1}{2}\tau_S^\frac{3}{2},
\qquad
S_c=\rho^\frac{1}{2}\Ckol^\frac{3}{4}\flux^\frac{1}{2}\tau_S^\frac{3}{2}
\quad
{\rm and}
\quad
S_i=\rho^{-\frac{3}{2}}\Ckol^\frac{3}{4}\flux^\frac{1}{2}\tau_S^\frac{3}{2}
\eqno(4.9)$$ From Eqn. (4.9), we identify the following sequence of ranges:

-   A large separation range $r>S$, in which the fluid velocity $\u^\smalP$ varies slowly on the scale of the relaxation time $\tau_S$.

-   A first intermediate range $S<r<S_c$ in which the fluid velocity $\u^\smalP$ is a fast variable, but still, $\tau_S$ is short compared with the crossing time of an eddy of size $r$; hence, Eqn. (4.2) has the form of a Langevin equation with a noise $\tau_S^{-1}\u^\smalP$ of constant amplitude on the scale of this crossing time. The cross-over scale $S$ will play an important role in the determination of the degree of non-ergodicity of the solid particle flow (see section VI).

-   A second intermediate range $S_c<r<S_i$, in which the crossing time is shorter than both the Stokes time and the eddy turn-over time, but is longer than the lifetime of an eddy of that size; hence, the solid particle moves ballistically with respect to the fluid.

-   A small 