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The idea of water potential still confuses me. I understand the general idea: water moves from an area of higher water potential to an area of lower water potential. However, I still do not understand the mechanism by which this occurs.

Take the example of putting salt on a slug. Putting salt on a slug lowers the dilute potential which in turn lowers the water potential on the outside of the slug. This lower water potential on the outside then draws water from the higher water potential of the inside of the slug.

I do not understand how this occurs. All the answers I find is that this just “occurs” because “nature wants to find equilibrium” (such as the second response on this Quora question). That makes sense in theory, but how does the cells of the slug “know” that there is now a lower water potential? What has changed?

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If I understand you correctly you are asking why osmosis happens the way it does. And obviously you want a more "real-life-approach" to this than the pure mathematically description via potentials.

Think about a cell filled with water that is divided by a permeable membrane that will only let water molecules pass through it. In this initial state the water molecules on both side will occasionally pass through the membrane switching sides. They do so by pure chance and as the particle-concentration on both sides are equal (there are 100% water particles on both sides) the pass through on both sides with an equal rate leaving the overall particle-distribution unchanged.
Now we go ahead and dilute salt in one half of the box. Leaving aside effects like hydration-hulls forming around the ions (which will magnify the effect described below) the effect of this dilution is that we now have a lower water-particle-concentration on that side. Therefore every now and then when a water particle would have been "thrown" at the membrane to pass through it it's not a water particle but an ion instead that can't pass the membrane. Therefore the rate at which the water-particles switch to the non-dilution-side is lower than the (unchanged) rate in which the water-particles flow into the dilution (by pure statistics). This leads to an increase in the overall particle count on the dilution-side.
This increase in particles also increase the pressure and therefore the rate at which random particles (on the dilution side) are being "thrown" at the membrane until this higher rate ultimately compensates the lower water-particle-concentration. At this point both sides are at an equilibrium again. However this equilibrium is in a state in which there are more water-particles on the dilution-side than there have been in the beginning.

And this is what we call osmosis.

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