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Obviously, if you drop materials into water near the surface of the Earth, they sort according to densities, with the least dense materials near the surface, and the most dense materials near the bottom. Common sense suggests this is due to gravity, because the more dense a material is, the more mass it has per unit of volume, and therefore, the more it interacts with any gravitational field. This should cause the most dense materials to experience a greater force due to gravity, causing them to sink more than others.

So my question then has two parts, which is, (1) is this the correct explanation for the apparent sorting of densities, and (2) if so, then what happens in space when you don't have any appreciable gravity?

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Yes that is more or less the correct explanation. The configuration with highest density materials at the bottom and lower density materials at the top is the one with the lowest potential energy, and thus the one favoured at equilibrium.

In space, far away from massive objects, there is no direction called “up” or “down”. Materials don’t sort themselves in order of decreasing density. To get a feel as to how they behaves, you can imagine what would happen if you filled a shallow plate with water, oil, rocks and so on, and look at their 2D movement from above. No direction in the plane of the fish is favoured and you get interesting shapes depending on happenstance and the dynamical properties of the materials.

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