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When I was a child, I read something like the following in a book whose name I have now forgotten (I tried to find the original quote on the web, but although there are some similar passages the precise source eludes me):

Imagine a solid sphere of iron the size of the Earth. Imagine, then, a fly that, once in a million years, sits down on that sphere, only to leave again the next moment. When the frictional heating from the fly's impact has caused the entirety of the sphere to evaporate, eternity has not yet started.

How long (ballpark estimate) would that actually take? Feel free to use any simplifying assumptions you like (Universe contains nothing but sphere and fly, sphere starts at uniform room temperature, indestructible, spherical fly that is at room temperature at each impact, impact speed 3 m/s etc).

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Ballpark (based on the iron starting out at 0 degrees Kelvin and melting at 1538 and the earth's radius of about 6000000 meters and the mass of a fly about 12 milligrams and velocity of a fly about 2 meters per second) (EDIT: Also based on the assumption of no radiative cooling of the sphere, i.e., perfect transfer of fly-bumping into heating the sphere and that heat is not dissipated into the larger universe, i.e., assume perfect thermal isolation of the Super Fly and the Iron Sphere) estimate:

About $10^{42}$ years.

That's a good long while. Given your views on the age of the universe and the likelihood of eternal recurrence, it may well be the case that eternity has started and stopped a few times by then. ;P

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    $\begingroup$ What about the radiative cooling of the sphere? $\endgroup$ – Kieran Hunt Feb 23 '15 at 9:22
  • $\begingroup$ That would change things. But it was not possible to include in the calculation because it would require knowledge of the contents of the rest of the universe. So, in my calculation the iron sphere should be considered to be alone in perfectly reflecting box with an invincible fly. $\endgroup$ – hft Feb 23 '15 at 9:27
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    $\begingroup$ "Feel free to use any simplifying assumptions you like... Universe contains nothing but sphere and fly" - I agree with your calculation, but it is worth stating that the effect of radiative cooling in this scenario would make the endpoint unreachable. $\endgroup$ – Kieran Hunt Feb 23 '15 at 9:29
  • $\begingroup$ That is correct. However, given that the Super Fly acts as an unlimited (though very slow-acting) input of energy to the system, and given a static finite universe, the iron and the entire universe would eventually reach the melting point of iron. Though, of course, in real life the universe may isn't static and there are no Super Flys. $\endgroup$ – hft Feb 23 '15 at 17:53

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