I am trying to figure out why vibrations (say, from an engine) loosen screws. It seems to me that there is evident symmetry between loosening and tightening a screw. I am wondering what breaks this symmetry.
The forces on the screw are not symmetric. Once the screw is no longer turning loosely in the hole tightening the screw compresses the two materials held together (i.e. increases the stress on the material, i.e. stores energy in the material), while loosening reduced the compression (i.e. releases the stress).
So a random dislocation will be more likely to occur in the "loose" direction than the "tight" direction.
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Regardless of whether the "local" situation is symmetric or not with respect to loosening and tightening, what you essentially have is a random walk. At any point in time, the screw can stay where it is, get a little looser or get a little tighter. There is, in practical terms, a limit as to how tight the screw can get but no limit on how loose. For any degree of looseness, there's a positive probability that you'll eventually reach that point, assuming the vibration is strong enough to move the screw at all.