See the Math Density Fluctuations and the RPA

The natural coordinates for describing emergent behavior in a quantum plasma or a metal are not the single particle coordinates, ri, but rather the fluctuations in the system density around its average value, ,

In these coordinates, the electrostatic interaction between electrons is seen to describe an interaction between density fluctuations, as it takes the simple form,

where i is not = j and k is not = 0. Assuming the usual form for the kinetic energy of the electrons, , it is easy to show that in the long wavelength limit, the equation of motion for the density fluctuation takes the form,

in either a classical or quantum calculation, if in determining that motion one keeps only the terms in the interaction that involve that same momentum q, i.e., neglects terms that involve products of density fluctuations

where q is not = k. In so doing, one is making the argument that such terms will be small because of the randomly varying phases of the terms in the exponential—hence the term, random phase approximation, or RPA.