From the continuity equation ##\frac{\partial \rho}{\partial t}+\rho (\nabla • u)=0## where ##\rho## is the mass density and is homogeneous and ##u## is the velocity of expansion or contraction.

For an expanding volume this becomes ##\nabla•u=\frac{\dot v(t)}{v(t)}=\Theta## which gives the rate of expansion ,##v(t)## is the volume.

Now if we consider our volume ##V(t)## to be consist of small patches of volume ##v(t)## where ##\rho## is homogeneous in ##v(t)## ,local expansion rate of volume ##V(t)## is therefore ##\Theta## ,now if the variance of this ##\Theta## is taken over ##V(t)## it will be ##\sigma_\Theta=[\bar{\Theta^2}-(\bar \Theta)^2]##.

Now my question is it small scale homogeneity is sufficient for this non-zero variance of volume expansion rate??

For an expanding volume this becomes ##\nabla•u=\frac{\dot v(t)}{v(t)}=\Theta## which gives the rate of expansion ,##v(t)## is the volume.

Now if we consider our volume ##V(t)## to be consist of small patches of volume ##v(t)## where ##\rho## is homogeneous in ##v(t)## ,local expansion rate of volume ##V(t)## is therefore ##\Theta## ,now if the variance of this ##\Theta## is taken over ##V(t)## it will be ##\sigma_\Theta=[\bar{\Theta^2}-(\bar \Theta)^2]##.

Now my question is it small scale homogeneity is sufficient for this non-zero variance of volume expansion rate??

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