The Fourier transform of (
) with an
initial distribution 
is
and has the formal solution
Equation () can be expressed in terms of
the complete basis defined by equations () as
Substituting this into equations (), () we see that
and so 
is going to be of the form
where 
is an lth order polynomial in
t, with rank l tensor coefficients.
For 
,
all the moments of x, and their time derivatives will become
vanishingly small. So, we have
thus establishing the existence of a hydrodynamic regime, where the time development of swarms is characterized by constant transport coefficients.
If the total number of particles is conserved, then 
at all times. This means from equation () that 
, so that 
is constant.
Since for large times, 
, this means
that 
at all times.
