Brownian motion of two interacting particles under a square-well potential

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Abstract
The Smoluchowski equation for the Brownian motion of two interacting particles through a square-well potential whose heights are infinitely large at the origin and finite at the other positions of u (>0) is solved exactly for the Laplace transform of the conditional density with respect to time t. The analytical expression for the distinct part of the dynamic structure factor at the initial time with the δ function has also been obtained exactly. Moreover, we have calculated the asymptotic behavior of the mean-square displacement expressed as an explicit function of t and found that it is a function of the height of the potential at u, which directly indicates a deviation from the Einstein relation.

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