Question regarding gradient of pressure term

Hey guys!

I am hoping you could help me a little bit with the maths here. I am looking at this formula:

And I just want to understand to understand how the expression for the pressure relationship between particle a and b is derived? Is it just some kind over averaging being applied, which I don't understand? (red line)

Secondly, any special reason to why we do not use the relationship which is used at the stress tensor (blue line) for the pressure gradient?

Could someone clarify it for me?

Kind regards, Ahmed
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