Brownian motion is collection of collision between any two particles.
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You do not need to know the interaction force in order to calculate the result after collision.

It is possible to have two collision with the same initial condition and the same outcome, however, the interaction force is different.

As a simple model: You can assume there is a spring action when  two particles collide with each other.
You are welcomed to check out [url=http://www.phy.ntnu.edu.tw/ntnujava/index.php?topic=193.0]Simulate 1D collision with a virtual spring[/url]