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When I take a panorama with a 80mm lens at F/2.0 on a fullframe body such that the resulting image has a equivalent field of view of a 50mm lens on the same body, what would then be the equivalent aperture I would need to use on the 50mm lens to get the same DOF as in my panorama?
Im guessing F/1.25. There I calculated the factor 1.6 = 80mm/50mm and divided 2.0 by it. If this is the right way to calculate the equivalent aperture, what is the physical reason behind that?
Depth of field changes with the resolution. If we disregard resolution and state that we want things blurred in the same manner assuming that resolution is infinite (and consequently depth of field is mathematically speaking 0 and not useful as a measure), the principal deciding criterion once we ascertain the same focusing distance is how large of an aperture we are looking out of.
80mm/2.0 is the same as 50mm/1.25 as you correctly determined. That makes for an "eye" (it's actually called "entrance pupil") of 40mm and means that background blur (at infinite distance) will have the same diameter that a disk of 40mm would have at the focusing distance.
With regard to foreground blur (at quite closer distance than the focusing distance), the blur disk is about 40mm in the world (it actually shrinks to zero as you get to the focusing distance and then grows again). So that kind of setup will be good at making a fence significantly closer than the focusing distance and with holes/pattern smaller than 40mm disappear on the photograph.
With regard to actual "depth of field", it all depends on what kind of blur circle you are prepared to call "in-focus" and that depends on viewing scale and/or sensor resolution and/or optical resolution. But if you are not looking at the detail level but at the big picture, the blur radius thing is what will be apparent and scale-independent and will define the equivalence of pictures/views.
