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My understanding of circuits which are not supplied an e.m.f. is that the electrons randomly just flow about in random directions, and since there's so many of them, probability dictates that any forwards or useful movement is cancelled almost perfectly by the negative or unwanted movement.

If this understanding is correct, is it theoretically possible for the electrons to, under an extremely low probability, mostly flow in the correct direction, and generate a current, perhaps lighting up a lamp? I'm talking even probabilities like 1x10^-1000, just wondering if it is at all possible.

This seems wrong to me and if anyone could explain why this is not possible, or the flaw in my logic, that would be greatly appreciated.

Apologies if there's some huge hole in my logic, or I'm being extremely stupid, I'm a student studying Physics below university level and thus know little more than basic outlines of Physical theories.

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    $\begingroup$ If you're "talking even probabilities like $1 : 10^{-1000}$" then the answer to pretty much any question you can think of approaches "yes". Read any Douglas Adams? (Which is not saying that everything he writes makes any sense scientifically.) Though you'll generally need numbers more like $10^{10^{20}}$ for "interesting macroscopic weirdness". $\endgroup$ – leftaroundabout Mar 31 '14 at 21:00
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    $\begingroup$ I'm more interested in if it's actually possible - I was expecting someone to notify me of a hole in my elementary current flow knowledge, if anything. - And I haven't! I really need to read Hitchhiker's at some point, though. $\endgroup$ – Ashley Davies Mar 31 '14 at 21:01
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    $\begingroup$ In a sense, the thermal motion of the atoms in the light bulb are already giving off light all the time even when it's turned off. It's not visible light, though. $\endgroup$ – endolith May 16 '14 at 3:13
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Yes, it is possible.

The simplest qualitative answer to this is that, at the microscopic level, the electrons in a conductor are dictated by quantum mechanics, which is inherently probabilistic. Velocities and positions are rarely ever totally excluded from a given value; it's just insanely unlikely for a single electron to attain that given value. Expecting it to happen for all of them makes it much, much worse. But it is possible.

A more classic example of this question is throwing a tennis ball against a brick wall: you can calculate how many times you'd have to throw it for all the atoms in the ball to tunnel through the potential of the wall, but you find that it's much more than the age of the universe.

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