Consider a particle under a force field that has a velocity parallel to the field direction and a speed proportional to the square of the magnitude of the electric field (any other non-linearity can be employed):
The effective mobility of the particle (the relationship between small changes in drift velocity  with respect to small changes in electric field ) can be expressed in Cartesian coordinates as:
Combining (5), (6) and (7) we get:
Further consider the field E is applied in a plane and it rotates counter-clockwise at angular frequency , such that the field components are:
Substituting (10) and (11) in (8) and (9) and simplifying using trigonometric identities results in a sum of constant terms, sine and cosine, at angular frequency . The next calculations will be performed  such that only the cosine terms at angular frequency  will yield non-zero net drift velocity - therefore we need only evaluate these terms, which will be abbreviated  and . The following is obtained:
Let  and  take the form of a small quadrupole field of intensity  that varies in a sinusoidal manner proportional to  such that:
Substituting (14) and (15) into (12) and (13) and taking the time average we obtain:
which can be summarized in vector notation to:
Equation (18) shows that for all positions  the time averaged velocity is in the direction toward the origin (concentrating the particles towards the origin), with speed proportional to the mobility coefficient k, the strength of the rotating field E and the strength of the perturbing quadrupole field .